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Frank Nielsen
Frédéric Barbaresco
(Eds.)
LNCS 9389
Second International Conference, GSI 2015
Palaiseau, France, October 28–30, 2015
Proceedings
Geometric Science
of Information
123

Lecture Notes in Computer Science 9389
Commenced Publication in 1973
Founding and Former Series Editors:
Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen
Editorial Board
David Hutchison
Lancaster University, Lancaster, UK
Takeo Kanade
Carnegie Mellon University, Pittsburgh, PA, USA
Josef Kittler
University of Surrey, Guildford, UK
Jon M. Kleinberg
Cornell University, Ithaca, NY, USA
Friedemann Mattern
ETH Zurich, Zürich, Switzerland
John C. Mitchell
Stanford University, Stanford, CA, USA
Moni Naor
Weizmann Institute of Science, Rehovot, Israel
C. Pandu Rangan
Indian Institute of Technology, Madras, India
Bernhard Steffen
TU Dortmund University, Dortmund, Germany
Demetri Terzopoulos
University of California, Los Angeles, CA, USA
Doug Tygar
University of California, Berkeley, CA, USA
Gerhard Weikum
Max Planck Institute for Informatics, Saarbrücken, Germany

More information about this series at http://www.springer.com/series/7412

Frank NielsenFrédéric Barbaresco (Eds.)
GeometricScience
ofInformation
Second International Conference, GSI 2015
Palaiseau, France, October 28–30, 2015
Proceedings
123

Editors
Frank Nielsen
École Polytechnique, LIX
Palaiseau
France
and
Sony Computer Science Laboratories, Inc.
Tokyo
Japan
Frédéric Barbaresco
Thales Land and Air Systems
Limours
France
ISSN 0302-9743 ISSN 1611-3349 (electronic)
Lecture Notes in Computer Science
ISBN 978-3-319-25039-7 ISBN 978-3-319-25040-3 (eBook)
DOI 10.1007/978-3-319-25040-3
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Preface
On behalf of both the Organizing and the Scientiﬁc Committees, it is our great pleasure
to welcome you to the proceedings of the Second International SEE Conference on
“Geometric Science of Information”(GSI 2015), hosted byÉcole Polytechnique
(Palaiseau, France), during October 28–30, 2015 (http://www.gsi2015.org/).
GSI 2015 beneﬁted from the scientiﬁc sponsorship of Sociétéde Mathématique
Appliquées et Industrielles (SMAI,smai.emath.fr/) and theﬁnancial sponsorship of:
–CNRS
–École Polytechnique
–Institut des Systèmes Complexes
–Inria (http://www.inria.fr/en/)
–Telecom ParisTech
–THALES (www.thalesgroup.com)
GSI 2015 was also supported by CNRS Federative Networks MIA and ISIS.
The 3-day conference was organized in the framework of the relations set up
between SEE (http://www.see.asso.fr/) and the following scientiﬁc institutions or
academic laboratories:École Polytechnique,École des Mines de Paris, INRIA,
Supélec, UniversitéParis-Sud, Institut Mathématique de Bordeaux, Sony Computer
Science Laboratories, Telecom SudParis, and Telecom ParisTech.
We would like to express our thanks to the Computer Science Department LIX of
École Polytechnique for hosting this second scientiﬁc event at the interface between
geometry, probability, and information geometry. In particular, we warmly thank
Evelyne Rayssac of LIX,École Polytechnique, for her kind administrative support that
helped us book the auditorium and various resources atÉcole Polytechnique, and
Olivier Bournez (LIX Director) for providingﬁnancial support.
The GSI conference cycle was initiated by the Brillouin Seminar Team (http://
repmus.ircam.fr/brillouin/home). The 2015 event was motivated in continuing theﬁrst
initiatives launched in 2013 (see LNCS proceedings 8085,http://www.springer.com/
us/book/9783642400193). We mention that in 2011, we organized an Indo-French
workshop on“Matrix Information Geometry”that yielded an edited book in 2013
(http://www.springer.com/us/book/9783642302312).
The technical program of GSI 2015 covered all the main topics and highlights in the
domain of“geometric science of information”including information geometry mani-
folds of structured data/information and their advanced applications. These proceedings
consist solely oforiginal research papersthat were carefullypeer-reviewedby two or
three experts and revised before acceptance.
The program included the renown invited speaker Professor Charles-Michel Marle
(UPMC, UniversitéPierre et Marie Curie, Paris, France), who gave a talk on“Actions
of Lie Groups and Lie Algebras on Symplectic and Poisson Manifolds,”and three
distinguished keynote speakers:

–Professor Marc Arnaudon (Bordeaux University, France):“Stocastic Euler-Poincaré
Reduction”
–Professor Tudor Ratiu (EPFL, Switzerland):“Symmetry Methods in Geometric
Mechanics”
–Professor Matilde Marcolli (Caltech, USA):“From Geometry and Physics to
Computational Linguistics”
A short course was given by Professor Dominique Spehner (Grenoble University,
France) on the“Geometry on the Set of Quantum States and Quantum Correlations”
chaired by Roger Balian (CEA, France).
The collection of papers have been arranged into the following 17 thematic sessions,
illustrating the richness and versatility of theﬁeld:
–Dimension Reduction on Riemannian Manifolds
–Optimal Transport
–Optimal Transport and Applications in Imagery/Statistics
–Shape Space and Diffeomorphic Mappings
–Random Geometry and Homology
–Hessian Information Geometry
–Topological Forms and Information
–Information Geometry Optimization
–Information Geometry in Image Analysis
–Divergence Geometry
–Optimization on Manifold
–Lie Groups and Geometric Mechanics/Thermodynamics
–Computational Information Geometry
–Lie Groups: Novel Statistical and Computational Frontiers
–Geometry of Time Series and Linear Dynamical Systems
–Bayesian and Information Geometry for Inverse Problems
–Probability Density Estimation
Historical Background
As for theﬁrst edition of GSI (2013) and in past publications (https://www.see.asso.fr/
node/11950), GSI 2015 addressed inter-relations between different mathematical
domains such as shape spaces (geometric statistics on manifolds and Lie groups,
deformations in shape space), probability/optimization and algorithms on manifolds
(structured matrix manifold, structured data/information), relational and discrete metric
spaces (graph metrics, distance geometry, relational analysis), computational and
Hessian information geometry, algebraic/inﬁnite dimensional/Banach information
manifolds, divergence geometry, tensor-valued morphology, optimal transport theory,
and manifold and topology learning, as well as applications such as geometries of
audio-processing, inverse problems, and signal processing.
At the turn of the century, new and fruitful interactions were discovered between
several branches of science: information science (information theory, digital commu-
nications, statistical signal processing), mathematics (group theory, geometry and
VI Preface

topology, probability, statistics), and physics (geometric mechanics, thermodynamics,
statistical physics, quantum mechanics).
From Statistics to Geometry
In the middle of the last century, a new branch in the geometric approach of statistical
problems was initiated independently by Harold Hotelling and Calyampudi Radhakr-
ishna Rao, who introduced a metric space in the parameter space of probability den-
sities. The metric tensor was proved to be equal to the Fisher information matrix. This
result was axiomatized by Nikolai Nikolaevich Chentsov in the framework of category
theory. This idea was also latent in the work of Maurice Fréchet, who had noticed that
the“distinguished densities”that reach lower bounds of statistical estimators are
deﬁned by a function that is given by a solution of the Legendre–Clairaut equation
(cornerstone equation of“information geometry”), and in the works of Jean-Louis
Koszul with a generalized notion of characteristic function.
From Probability to Geometry
Probability is again the subject of a new foundation to apprehend new structures and
generalize the theory to more abstract spaces (metric spaces, shape space, homoge-
neous manifolds, graphs). An initial attempt to probability generalization in metric
spaces was made by Maurice Fréchet in the middle of the last century, in the frame-
work of abstract spaces topologically afﬁne and“distance space”(“espace distancié”).
More recently, Misha Gromov, at IHES (Institute of Advanced Scientiﬁc Studies),
indicated the possibilities for (non-)homological linearization of basic notions of
probability theory and also the replacement of real numbers as values of probabilities
by objects of suitable combinatorial categories. In parallel, Daniel Bennequin, from
Institut mathématique de Jussieu, observed that entropy is a universal co-homological
class in a theory associated with a family of observable quantities and a family of
probability distributions.
From Groups Theory to Geometry
As observed by Gaston Bachelard,“The group provides evidence of a mathematic
closed on itself. Its discovery closes the era of conventions, more or less independent,
more or less coherent.”About Elie Cartan’s work on group theory, Henri Poincarésaid
that“The problems addressed by Elie Cartan are among the most important, most
abstract, and most general dealing with mathematics; group theory is, so to speak, the
whole mathematics, stripped of its material and reduced to pure form. This extreme
level of abstraction has probably made my presentation a little dry. To assess each
of the results, I would have had to virtually render it the material of which it had been
stripped; but this refund can be made in a thousand different ways; and this is the only
Preface VII

form that can be found as well as a host of various garments, which is the common link
between mathematical theories whose proximity is often surprising.”
From Mechanics to Geometry
The last elaboration of geometric structure on information is emerging at the
inter-relations between“geometric mechanics”and“information theory”that was
largely debated at the GSI 2015 conference with invited speakers including
C.M. Marle, T. Ratiu, and M. Arnaudon. Elie Cartan, the master of geometry during the
last century, said:“distinguished service that has rendered and will make even the
absolute differential calculus of Ricci and Levi–Civita should not prevent us from
avoiding too exclusively formal calculations, where debauchery indices often mask a
very simple geometric fact. It is this reality that I have sought to put in evidence
everywhere.”Elie Cartan was the son of Joseph Cartan, who was the village black-
smith, and Elie recalled that his childhood had passed under“blows of the anvil, which
started every morning from dawn.”One can imagine that the hammer blows made by
Joseph on the anvil, giving shape and curvature to the metal, inﬂuenced Elie’s mind
with germinal intuition of fundamental geometric concepts. The alliance between
geometry and mechanics is beautifully illustrated by the image of Forge, in the painting
of Velasquez about the Vulcan God (see Figure1). This concordance of meaning is
also conﬁrmed by the etymology of the word“forge,”which comes from late four-
teenth century,“a smithy,”from the Old Frenchforge“forge, smithy”(twelfth cen-
tury), earlierfaverge, from the Latinfabrica“workshop, smith’s shop,”from faber
(genitive fabri)“workman in hard materials, smith.”
Fig. 1.Into the Flaming Forge of Vulcan, into the Ninth Sphere, Mars descends in order to retemper
hisﬂaming sword and conquer the heart of Venus (Diego Velázquez, Museo Nacional del Prado).
Public domain image, courtesy ofhttps://en.wikipedia.org/wiki/Apollo_in_the_Forge_of_Vulcan
VIII Preface

As Henri Bergson said in his bookThe Creative Evolutionin 1907:“As regards
human intelligence, there is not enough [acknowledgment] that mechanical invention
wasﬁrst its essential approach…we should say perhaps notHomo sapiens, butHomo
faber. In short, intelligence, considered in what seems to be its original feature, is the
faculty of manufacturing artiﬁcial objects, especially tools to make tools, and of
indeﬁnitely varying the manufacture.”
Geometric Science of Information: A new Grammar of Sciences
Henri Poincarésaid that“mathematics is the art of giving the same name to different
things”(“La mathématique est l’art de donner le même nomàdes choses différentes”in
Science et méthode, 1908). By paraphrasing Henri Poincaré, we could claim that the
“geometric science of information”is the art of giving the same name to different
sciences. The rules and the structures developed at the GSI 2015 conference comprise a
kind of new grammar for these sciences.
We give our thanks to all the authors and co-authors for their tremendous effort and
scientiﬁc contribution. We would also like to acknowledge all the Organizing and
Scientiﬁc Committee members for their hard work in evaluating the submissions. We
warmly thank Jean Vieille, Valerie Alidor, and Flore Manier from the SEE for their
kind support.
Preface IX

As with GSI 2013, a selected number of contributions focusing on a core topic were
invited to contribute a chapter without page restriction to the edited bookGeometric
Theory of Information(http://www.springer.com/us/book/9783319053165) in 2014.
Similarly, for GSI 2015, we invite prospective authors to submit their original work to
a special issue on“advances in differential geometrical theory of statistics”of the
MDPIEntropyjournal (http://www.mdpi.com/journal/entropy/special_issues/entropy-
statistics).
It is our hope that theﬁne collection of peer-reviewed papers presented in these
LNCS proceedings will be a valuable resource for researchers working in theﬁeld of
information geometry and for graduate students.
July 2015 Frank Nielsen
Frédéric Barbaresco
X Preface

Organization
Program Chairs
Frédéric Barbaresco Thales Air Systems, France
Frank Nielsen École Polytechnique, France and Sony CSL, Japan
Scientiﬁc Committee
Pierre-Antoine Absil University of Louvain, Belgium
Bijan Afsari John Hopkins University, USA
Stéphanie Allassonnière École Polytechnique, France
Jésus Angulo MINES ParisTech, France
Marc Arnaudon Institut de Math ématiques de Bordeaux, France
Michael Aupetit Qatar Computing Research Institute, Quatar
Roger Balian Academy of Sciences, France
Barbara Trivellato Politecnico di Torino, Italy
Pierre Baudot Max Planck Institute of Leipzig, Germany
Daniel Bennequin University of Paris Diderot, France
Yannick Berthoumieu École Nationale d’Electronique, Informatique et
Radiocommunications de Bordeaux, France
Jérémie Bigot Institut de Math ématiques de Bordeaux, France
Silvère Bonnabel Mines ParisTech, France
Michel Boyom University of Montpellier, France
Michel Broniatowski University of Pierre and Marie Curie, France
Martins Bruveris Brunel University London, UK
Charles Cavalcante Universidade Federal do Ceara, Brazil
Frédéric Chazal Inria, France
Arshia Cont IRCAM, France
Gery de Saxcé Universitédes Sciences et des Technologies de Lille,
France
Laurent Decreusefond Telecom ParisTech, France
Michel Deza ENS Paris, France
Stanley Durrleman Inria, France
Patrizio Frosini University of Bologna, Italy
Alfred Galichon New York University, USA
Alexander Ivanov Imperial College, UK
Jérémie Jakubowicz Institut Mines-Telécom, France
Hongvan Le Mathematical Institute of ASCR, Czech Republik
Nicolas Le Bihan University of Melbourne, Australia
Luigi Malagò Shinshu University, Japan

Jonathan Manton University of Melbourne, Australia
Jean-François
Marcotorchino
Thales, France
Bertrand Maury Universit éParis-Sud, France
Ali Mohammad-Djafari Supelec, France
Richard Nock NICTA, Australia
Yann Ollivier CNRS, France
Xavier Pennec Inria, France
Michel Petitjean Universit éParis 7, France
Gabriel Peyré CNRS, France
Giovanni Pistone Collegio Carlo Alberto de Castro Statistics Initiative,
Italy
Olivier Rioul T élécom ParisTech, France
Said Salem IMS Bordeaux, France
Olivier Schwander University of Geneva, Switzerland
Rodolphe Sepulchre Cambridge University, UK
Hichem Snoussi University of Technology of Troyes, France
Alain Trouvé ENS Cachan, France
Claude Vallée Poitiers University, France
Geert Verdoolaege Ghent University, Belgium
Rui Vigelis Federal University of Ceara, Brazil
Susan Holmes Stanford University, USA
Martin Kleinsteuber Technische UniversitätMünchen, Germany
Shiro Ikeda ISM, Japan
Martin Bauer University of Vienna, Austria
Charles-Michel Marle UniversitéPierre et Marie Curie, France
Mathilde Marcolli Caltech, USA
Jean-Philippe Ovarlez Onera, France
Jean-Philippe Vert Mines ParisTech, France
Allessandro Sarti École des hautesétudes en sciences sociales, Paris
Jean-Paul Gauthier University of Toulon, France
Wen Huang University of Louvain, Belgium
Antonin Chambolle École Polytechnique, France
Jean-François Bercher ESIEE, France
Bruno Pelletier University of Rennes, France
Stephan Weis Universidade Estadual de Campinas, Brazil
Gilles Celeux Inria, France
Jean-Michel Loubes Toulouse University, France
Anuj Srivastana Florida State University, USA
Johannes Rauh Leibniz Universit ät Hannover, Germany
Joan Alexis Glaunes Mines ParisTech, France
Quentin Mérigot Universit éParis-Dauphine, Paris
K.S. Subrahamanian
Moosath
University of Calicut, India
K.V. Harsha Indian Institute of Space Science and Technology,
India
XII Organization

Emmanuel Trelat UPMC, France
Lionel Bombrun IMS Bordeaux, France
Olivier Cappé Telecom Paris, France
Stephan Huckemann Institut f ür Mathematische Stochastik; Göttingen,
Germany
Piotr Graczyk University of Angers, France
Fernand Meyer Mines ParisTech
Corinne Vachier Universit éParis Est Créteil, France
Tudor Ratiu EPFL, Swiss
Klas Modin Chalmers University of Technology, G öteborg,
Sweden
HervéLombaert Inria, France
Michèle Basseville IRISA, France
Juliette Matiolli Thales, France
Peter D. Grünwald CWI, Amsterdam, The Netherlands
François-Xavier Viallard CEREMADE, Paris, France
Guido Francisco Montúfar Max Planck Institute for Mathematics in the Sciences,
Leipzig, Germany
Emmanuel Chevallier Mines ParisTech, France
Christian Leonard École Polytechnique, France
Nikolaus Hansen Inria, France
Laurent Younes John Hopkins University, USA
Sylvain Arguillère John Hopkins University, USA
Shun-Ichi Amari RIKEN, Japan
Julien Rabin ENSICAEN, France
Dena Asta Carnegie Mellon University, USA
Pierre-Yves GousenbourgerÉcole Polytechnique de Louvain, Belgium
Nicolas Boumal Inria and ENS Paris, France
Jun Zhang University of Michigan, Ann Arbor, USA
Jan Naudts University of Antwerp, Belgium
Alexis Decurninge Huawei Technologies, Paris
Roman Belavkin Middlesex University, UK
Hugo Boscain École Polytecnique, France
Eric Moulines Telecom ParisTech, France
Udo Von Toussaint Max-Planck-Institut fuer Plasmaphysik, Garching,
Germany
Jean-Philippe Anker University of Orléans, France
Charles Bouveyron University Paris Descartes, France
Michael Blum IMAG, France
Sylvain Chevallier IUT of V élizy, France
Jeremy Bensadon LRI, France
Philippe Cuvillier IRCAM, France
Frédéric Barbaresco Thales, France
Frank Nielsen École Polytechnique, France and Sony CSL, Japan
Organization XIII

Sponsors and Organizer
XIV Organization

Contents
Dimension Reduction on Riemannian Manifolds
Evolution Equations with Anisotropic Distributions and Diffusion PCA . . . . . 3
Stefan Sommer
Barycentric Subspaces and Affine Spans in Manifolds . . . . . . . . . . . . . . . . . 12
Xavier Pennec
Dimension Reduction on Polyspheres with Application to Skeletal
Representations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Benjamin Eltzner, Sungkyu Jung, and Stephan Huckemann
Affine-Invariant Riemannian Distance between Infinite-Dimensional
Covariance Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
HàQuang Minh
A Sub-Riemannian Modular Approach for Diffeomorphic Deformations . . . . 39
Barbara Gris, Stanley Durrleman, and Alain Trouvé
Optimal Transport
The Nonlinear Bernstein-Schrödinger Equation in Economics . . . . . . . . . . . . 51
Alfred Galichon, Scott Duke Kominers, and Simon Weber
Some Geometric Consequences of the Schrödinger Problem . . . . . . . . . . . . . 60
Christian Léonard
Optimal Transport, Independance Versus Indetermination Duality, Impact
on a New Copula Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Benoit Huyot, Yves Mabiala, and J.F. Marcotorchino
Optimal Mass Transport over Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Yongxin Chen, Tryphon Georgiou, and Michele Pavon
Optimal Transport and Applications in Imagery/Statistics
Non-convex Relaxation of Optimal Transport for Color Transfer
between Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Julien Rabin and Nicolas Papadakis

Generalized Pareto Distributions, Image Statistics and Autofocusing in
Automated Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Reiner Lenz
Barycenter in Wasserstein Spaces: Existence and Consistency. . . . . . . . . . . . 104
Thibaut Le Gouic and Jean-Michel Loubes
Multivariate L-Moments Based on Transports. . . . . . . . . . . . . . . . . . . . . . . 109
Alexis Decurninge
Shape Space and Diffeomorphic Mappings
Spherical Parameterization for Genus Zero Surfaces Using
Laplace-Beltrami Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Julien Lefèvre and Guillaume Auzias
Biased Estimators on Quotient Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Nina Miolane and Xavier Pennec
Reparameterization Invariant Metric on the Space of Curves. . . . . . . . . . . . . 140
Alice Le Brigant, Marc Arnaudon, and Frédéric Barbaresco
Invariant Geometric Structures on Statistical Models . . . . . . . . . . . . . . . . . . 150
Lorenz Schwachhöfer, Nihat Ay, Jürgen Jost, and Hông VânLê
The Abstract Setting for Shape Deformation Analysis and LDDMM
Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Sylvain Arguillère
Random Geometry/Homology
The Extremal Index for a Random Tessellation. . . . . . . . . . . . . . . . . . . . . . 171
Nicolas Chenavier
A Two-Color Interacting Random Balls Model for Co-localization Analysis
of Proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
F. Lavancier and C. Kervrann
Asymptotics of Superposition of Point Processes. . . . . . . . . . . . . . . . . . . . . 187
L. Decreusefond and A. Vasseur
Asymptotic Properties of Random Polytopes . . . . . . . . . . . . . . . . . . . . . . . 195
Pierre Calka
Asymmetric Topologies on Statistical Manifolds . . . . . . . . . . . . . . . . . . . . . 203
Roman V. Belavkin
XVI Contents

Hessian Information Geometry
Hessian Structures and Non-invariant (F,G)-Geometry on a Deformed
Exponential Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
K.V. Harsha and K.S. Subrahamanian Moosath
New Metric and Connections in Statistical Manifolds . . . . . . . . . . . . . . . . . 222
Rui F. Vigelis, David C. de Souza, and Charles C. Cavalcante
Curvatures of Statistical Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
Barbara Opozda
The Pontryagin Forms of Hessian Manifolds . . . . . . . . . . . . . . . . . . . . . . . 240
J. Armstrong and S. Amari
Matrix Realization of a Homogeneous Cone. . . . . . . . . . . . . . . . . . . . . . . . 248
Hideyuki Ishi
Multiply CR-Warped Product Statistical Submanifolds of a Holomorphic
Statistical Space Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Michel Nguiffo Boyom, Mohammed Jamali,
and Mohammad Hasan Shahid
Topological Forms and Information
Information Algebras and Their Applications . . . . . . . . . . . . . . . . . . . . . . . 271
Matilde Marcolli
Finite Polylogarithms, Their Multiple Analogues and the Shannon
Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Philippe Elbaz-Vincent and Herbert Gangl
Heights of Toric Varieties, Entropy and Integration over Polytopes . . . . . . . . 286
JoséIgnacio Burgos Gil, Patrice Philippon, and Martín Sombra
Characterization and Estimation of the Variations of a Random Convex Set
by Its Meann-Variogram: Application to the Boolean Model . . . . . . . . . . . . 296
Saïd Rahmani, Jean-Charles Pinoli, and Johan Debayle
Information Geometry Optimization
Laplace’s Rule of Succession in Information Geometry . . . . . . . . . . . . . . . . 311
Yann Ollivier
Standard Divergence in Manifold of Dual Affine Connections . . . . . . . . . . . 320
Shun-ichi Amari and Nihat Ay
Contents XVII

Transformations and Coupling Relations for Affine Connections. . . . . . . . . . 326
James Tao and Jun Zhang
Onlinek-MLE for Mixture Modeling with Exponential Families . . . . . . . . . . 340
Christophe Saint-Jean and Frank Nielsen
Second-Order Optimization over the Multivariate Gaussian Distribution. . . . . 349
Luigi Malagòand Giovanni Pistone
The Information Geometry of Mirror Descent . . . . . . . . . . . . . . . . . . . . . . . 359
Garvesh Raskutti and Sayan Mukherjee
Information Geometry in Image Analysis
Texture Classification Using Rao’s Distance on the Space of Covariance
Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Salem Said, Lionel Bombrun, and Yannick Berthoumieu
Color Texture Discrimination Using the Principal Geodesic Distance
on a Multivariate Generalized Gaussian Manifold . . . . . . . . . . . . . . . . . . . . 379
Geert Verdoolaege and Aqsa Shabbir
Bag-of-Components: An Online Algorithm for Batch Learning of Mixture
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
Olivier Schwander and Frank Nielsen
Statistical Gaussian Model of Image Regions in Stochastic Watershed
Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
Jesús Angulo
Quantization of Hyperspectral Image Manifold Using Probabilistic
Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
Gianni Franchi and Jesús Angulo
Divergence Geometry
Generalized EM Algorithms for Minimum Divergence Estimation. . . . . . . . . 417
Diaa Al Mohamad and Michel Broniatowski
Extension of Information Geometry to Non-statistical Systems: Some
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
Jan Naudts and Ben Anthonis
An Information Geometry Problem in Mathematical Finance . . . . . . . . . . . . 435
Imre Csiszár and Thomas Breuer
XVIII Contents

Multivariate Divergences with Application in Multisample Density Ratio
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
Amor Keziou
Generalized Mutual-Information Based Independence Tests . . . . . . . . . . . . . 454
Amor Keziou and Philippe Regnault
Optimization on Manifold
Riemannian Trust Regions with Finite-Difference Hessian Approximations
are Globally Convergent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
Nicolas Boumal
Block-Jacobi Methods with Newton-Steps and Non-unitary Joint Matrix
Diagonalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
Martin Kleinsteuber and Hao Shen
Weakly Correlated Sparse Components with Nearly Orthonormal
Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
Matthieu Genicot, Wen Huang, and Nickolay T. Trendafilov
Fitting Smooth Paths on Riemannian Manifolds: Endometrial Surface
Reconstruction and Preoperative MRI-Based Navigation . . . . . . . . . . . . . . . 491
Antoine Arnould, Pierre-Yves Gousenbourger, Chafik Samir,
Pierre-Antoine Absil, and Michel Canis
PDE Constrained Shape Optimization as Optimization on Shape
Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
Volker H. Schulz, Martin Siebenborn, and Kathrin Welker
Lie Groups and Geometric Mechanics/Thermodynamics
PoincaréEquations for Cosserat Shells: Application to Cephalopod
Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
Frederic Boyer and Federico Renda
Entropy and Structure of the Thermodynamical Systems . . . . . . . . . . . . . . . 519
Géry de Saxcé
Symplectic Structure of Information Geometry: Fisher Metric
and Euler-PoincaréEquation of Souriau Lie Group Thermodynamics. . . . . . . 529
Frédéric Barbaresco
Pontryagin Calculus in Riemannian Geometry . . . . . . . . . . . . . . . . . . . . . . 541
François Dubois, Danielle Fortuné, Juan Antonio Rojas Quintero,
and Claude Vallée
Contents XIX

Rolling Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
Krzysztof A. Krakowski, Luís Machado, and Fátima Silva Leite
Enlargement, Geodesics, and Collectives . . . . . . . . . . . . . . . . . . . . . . . . . . 558
Eric W. Justh and P.S. Krishnaprasad
Computational Information Geometry
Geometry of Goodness-of-Fit Testing in High Dimensional Low Sample
Size Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
Paul Marriott, Radka Sabolova, Germain Van Bever,
and Frank Critchley
Computing Boundaries in Local Mixture Models . . . . . . . . . . . . . . . . . . . . 577
Vahed Maroufy and Paul Marriott
Approximating Covering and Minimum Enclosing Balls in Hyperbolic
Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
Frank Nielsen and Gaëtan Hadjeres
From Euclidean to Riemannian Means: Information Geometry for SSVEP
Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
Emmanuel K. Kalunga, Sylvain Chevallier, Quentin Barthélemy,
Karim Djouani, Yskandar Hamam, and Eric Monacelli
Group Theoretical Study on Geodesics for the Elliptical Models . . . . . . . . . . 605
Hiroto Inoue
Path Connectedness on a Space of Probability Density Functions . . . . . . . . . 615
Shinto Eguchi and Osamu Komori
Lie Groups: Novel Statistical and Computational Frontiers
Image Processing in the Semidiscrete Group of Rototranslations . . . . . . . . . . 627
Dario Prandi, Ugo Boscain, and Jean-Paul Gauthier
Universal, Non-asymptotic Confidence Sets for Circular Means . . . . . . . . . . 635
Thomas Hotz, Florian Kelma, and Johannes Wieditz
A Methodology for Deblurring and Recovering Conformational States
of Biomolecular Complexes from Single Particle Electron Microscopy. . . . . . 643
Bijan Afsari and Gregory S. Chirikjian
Nonlinear Operators on Graphs via Stacks . . . . . . . . . . . . . . . . . . . . . . . . . 654
Santiago Velasco-Forero and Jesús Angulo
An Intrinsic Cramér-Rao Bound on Lie Groups . . . . . . . . . . . . . . . . . . . . . 664
Silvère Bonnabel and Axel Barrau
XX Contents

Geometry of Time Series and Linear Dynamical systems
Clustering Random Walk Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
Gautier Marti, Frank Nielsen, Philippe Very, and Philippe Donnat
A Common Symmetrization Framework for Iterative (Linear) Maps . . . . . . . 685
Alain Sarlette
New Model Search for Nonlinear Recursive Models, Regressions
and Autoregressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
Anna-Lena Kißlinger and Wolfgang Stummer
Random Pairwise Gossip onCATðjÞMetric Spaces . . . . . . . . . . . . . . . . . . 702
Anass Bellachehab and Jérémie Jakubowicz
Bayesian and Information Geometry for Inverse Problems
Stochastic PDE Projection on Manifolds: Assumed-Density and Galerkin
Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713
John Armstrong and Damiano Brigo
Variational Bayesian Approximation Method for Classification
and Clustering with a Mixture of Student-t Model. . . . . . . . . . . . . . . . . . . . 723
Ali Mohammad-Djafari
Geometric Properties of Textile Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732
Tomonari Sei and Ushio Tanaka
A Generalization of Independence and Multivariate Student’s
t-distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740
Monta Sakamoto and Hiroshi Matsuzoe
Probability Density Estimation
Probability Density Estimation on the Hyperbolic Space Applied to Radar
Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
Emmanuel Chevallier, Frédéric Barbaresco, and Jesús Angulo
Histograms of Images Valued in the Manifold of Colours Endowed
with Perceptual Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762
Emmanuel Chevallier, Ivar Farup, and Jesús Angulo
Entropy Minimizing Curves with Application to Automated Flight Path
Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770
Stephane Puechmorel and Florence Nicol
Contents XXI

Kernel Density Estimation on Symmetric Spaces. . . . . . . . . . . . . . . . . . . . . 779
Dena Marie Asta
Author Index............................................ 789
XXII Contents

Dimension Reduction on Riemannian
Manifolds

Evolution Equations with Anisotropic
Distributions and Diﬀusion PCA
Stefan Sommer
(B)
Department of Computer Science, University of Copenhagen, Copenhagen, Denmark
[email protected]
Abstract.This paper presents derivations of evolution equations for
the family of paths that in the Diﬀusion PCA framework are used for
approximating data likelihood. The paths that are formally interpreted
as most probable paths generalize geodesics in extremizing an energy
functional on the space of diﬀerentiable curves on a manifold with con-
nection. We discuss how the paths arise as projections of geodesics for
a (non bracket-generating) sub-Riemannian metric on the frame bundle.
Evolution equations in coordinates for both metric and cometric formula-
tions of the sub-Riemannian geometry are derived. We furthermore show
how rank-deﬁcient metrics can be mixed with an underlying Riemannian
metric, and we use the construction to show how the evolution equations
can be implemented on ﬁnite dimensional LDDMM landmark manifolds.
1 Introduction
The diﬀusion PCA framework [1,2] models data on non-linear manifolds as
samples from distributions generated by anisotropic diﬀusion processes. These
processes are mapped from Euclidean space to the manifold by stochastic devel-
opment in the frame bundle [3]. The construction is connected to a (non bracket-
generating) sub-Riemannian metric on the bundle of linear frames of the
manifold, the frame bundle.
Velocity vectors and length of geodesics are conventionally used for estimation
and statistics in Riemannian manifolds, i.e. for Principal Geodesic Analysis [4]or
tangent space statistics [5]. In contrast to this, the anisotropic nature of the dis-
tributions considered for Diﬀusion PCA makes geodesics for the sub-Riemannian
metric the natural vehicle for estimation and statistics. These paths were pre-
sented in [2] and formally interpreted as most probable paths for the driving dif-
fusion processes that are mapped fromR
n
toMby stochastic development.
In this paper, we present derivations of the evolution equations for the paths.
We discuss the role of frames as representing either metrics or cometrics and
how the sub-Riemannian metric is related to the Sasaki-Mok metric onFM.
We then develop a construction that allows the sub-Riemannian metric to be
deﬁned as a sum of a rank-deﬁcient generator and an underlying Riemannian
metric. Finally, we show how the evolution equations manifest themselves in a
speciﬁc case, the ﬁnite dimensional manifolds arising in the LDDMM landmark
matching problem.
cﬀSpringer International Publishing Switzerland 2015
F. Nielsen and F. Barbaresco (Eds.): GSI 2015, LNCS 9389, pp. 3–11, 2015.
DOI: 10.1007/978-3-319-25040-3
1

4 S. Sommer
1.1 Diﬀusion PCA
Diﬀusion PCA (DPCA, [1,2]) provides a generalization of the Euclidean Prin-
cipal Component Analysis (PCA) procedure to Riemannian manifolds or, more
generally, diﬀerentiable manifolds with connection. In contrast to procedures
such as Principal Geodesic Analysis (PGA, [4]), Geodesic PCA (GPCA, [6]), and
Horizontal Component Analysis (HCA, [7]), DPCA does not employ explicit rep-
resentations of low-dimensional subspaces. Instead of generalizing the maximum
variance/minimum residual formulation of PCA, it is based on a formulation
of PCA as a maximum likelihood ﬁt of a Gaussian distribution to data [8,9].
Through the process of stochastic development [3], a class of anisotropic distrib-
utions are deﬁned that generalizes normal distributions to the manifold situation.
DPCA is thereby a maximum likelihood ﬁt in this family of distributions.
2 Anisotropic Diﬀusions, Frame Bundles and
Development
Development and stochastic development provides an invertible mapϕ
(x,Xα)
from paths inR
n
starting at the origin to paths on the manifoldMstarting at
a given pointx∈M. The development mapϕ
(x,Xα)is dependent on both the
starting pointxand a frameX
αforT xM. Throughϕ
(x,Xα), diﬀusion processes
inR
n
map to processes onM. This construction is called the Eells-Elworthy-
Malliavin construction of Brownian motion [10]. We here outline the process of
development and stochastic development and describe its use in Diﬀusion PCA.
Let (M,∇) be a diﬀerentiable manifold of dimensionnwith connection∇.
For each pointx∈M,letF
xMbe the set of framesX α, i.e. ordered bases of
T
xM. The set{F xM}x∈Mcan be given a natural diﬀerential structure as a ﬁber
bundle onMcalled the frame bundleFM. It can equivalently be deﬁned as the
principal bundle GL(R
n
,TM). We let the mapπ FM:FM→Mdenote the
canonical projection. For a diﬀerentiable curvex
tinMwithx=x 0,aframe
X
α=X α,0forT x0
Mcan be parallel transported alongx tthus giving a path
(x
t,Xα,t)inFM. Such paths are called horizontal, and their derivatives form
n-dimensional subspaces of then+n
2
-dimensional tangent spacesT
(x,Xα)FM.
This horizontal subspaceHFMand the vertical subspaceVFMof vectors tan-
gent to the ﬁbersπ
−1
(x) together split the tangent spaces, i.e.T
(x,Xα)FM=
H
(x,Xα)FM⊕V
(x,Xα)FM. The split induces a mapπ ∗:HFM→TMand
isomorphismsπ
∗,(x,X α):H
(x,Xα)FM→T xMwith inversesπ
∗
(x,X
α)
, see Fig.1.
Usingπ
∗
(x,X
α)
, horizontal vector ﬁeldsH eonFMare deﬁned for vectorse∈R
n
byH e(u)=(ue)
∗
. In particular, the standard basis (e 1,...,en)onR
n
givesn
globally deﬁned horizontal vector ﬁeldsH
i∈HFM,i=1,...,nbyH i=H ei
.
Ahorizontal liftofx
tis a curve inFMtangent toHFMthat projects tox t.
Horizontal lifts are unique up to the choice of initial frameX
α,0.
LetW
tbe anR
n
valued semimartingale. A solution to the stochastic diﬀeren-
tial equationdU
t=
ﬀ
d
i=1
Hi(Ut)◦dW
i
t
inFMis called a stochastic development
ofW
t. The solution projects to a stochastic developmentX t=πFM(Ut)inM.

Evolution Equations with Anisotropic Distributions and Diﬀusion PCA 5
TFM
T
∗
FM
HFM FM
TMT
∗
M M
h+v∂ →h
π∗ πFM
πTM
gFM
gR
Fig. 1.Commutative diagram for the manifold, frame bundle, the horizontal subspace
HFMofTFM, a Riemannian metricg
Rand the sub-Riemannian metricg FMdeﬁned
below. The connection provides the splittingTFM=HFM⊕VFM. The restrictions
π
∗|H
(x,Xα)
Mare invertible mapsH (x,Xα)M→T xM.
We call the processW tinR
n
that throughϕmaps toX tfor the driving process
ofX
t. Since a normal distributionW∼N(0,Σ) can be obtained as the transi-
tion probability of a diﬀusion processW
tstopped at e.g.t= 1, a general class
of distributions on the manifoldMcan be deﬁned by stochastic development of
processesW
tresulting in random variablesX=X 1.
Diﬀusion PCA uses the map
ﬃ
Diﬀ
:FM→Dens(M) that for each point
(x, X
α)∈FMsends a Brownian motion inR
n
to a distributionX 1by starting
a diﬀusionU
tat (x, X α) and lettingX 1=πFM(U1) after normalization. The
pair (x, X
α) is analogous to the parameters (μ, Σ) for a Euclidean normal dis-
tribution: the pointx∈Mrepresents the starting point of the diﬀusion, and
X
αrepresents the square root covarianceΣ
1/2
. Diﬀusion PCA ﬁts distributions
obtained through
ﬃ
Diﬀ
by maximum likelihood to observed data, i.e. it optimizes
for the most probable parameters (x, X
α) for the anisotropic diﬀusion process.
3 Evolution Equations
For a Euclidean stationary driftless diﬀusion process with stochastic generator
Σ, the log-probability of a sample path can formally be written
ln ˜p
Σ(xt)∝−
Σ
1
0
˙xt
2
Σ
dt+c Σ (1)
with the norm·
Σgiven by the inner productv, w Σ=
ξ
Σ
−1/2
v, Σ
−1/2
w
ζ
.
Though only formal as the sample paths are almost surely nowhere diﬀerentiable,
the interpretation can be given a precise meaning by taking limits of piecewise
linear curves [11]. Turning to the manifold situation with the processes mapped
toMby stochastic development, the probability of observing a path can either
be deﬁned in the manifold by taking limits of small tubes around the curve,
or inR
n
trough its anti-development. With the former formulation, a scalar
curvature correction term must be added to (1) giving the Onsager-Machlup
functional ([12]). The latter formulation corresponds to ﬁnding probabilities of
paths for the driving processW
t. Taking the maximum of (1) gives geodesics as
most probable paths for the driving process whenΣis unitary.

6 S. Sommer
Let now (x t,Xα,t) be a path inFM. Recall that in DPCA,X α,trepresents the
square root covarianceΣ
1/2
atxt. SinceX α,tbeing a basis deﬁnes an invertible
mapR
n
→T xt
M, the norm· Σhas a direct analogue in the norm· Xα,t
deﬁned by the inner product
v, w
Xα,t
=
ξ
X
−1
α,t
v, X
−1
α,t
w
ζ
R
n
(2)
for vectorsv, w∈T
xt
M. The transport of the frame along paths in eﬀect deﬁnes
a transport of inner product along sample paths: the paths carry with them the
inner product deﬁned by the square root covarianceX
α,0atx0.
The inner product can equivalently be deﬁned as a metricg
Xα
:T
∗
x
M→
T
xM. Again using thatX α,tcan be considered a mapR
n
→Txt
,gXα
is deﬁned
byξ∂ →X
α((ξ◦X α)
Σ
) whereζis the standard identiﬁcation (R
n
)
∗
→R
n
.The
sequence of mappings deﬁningg
Xα
is illustrated below:
T
∗
x
t
M→(R
n
)
∗
→ R
n
→ T xt
M
ξ∂ →ξ◦X
α∂ →(ξ◦X α)
Σ
∂ →X α(ξ◦X α)
Σ
.
(3)
This deﬁnition uses theR
n
inner product in the deﬁnition ofζ. Its inverse gives
the cometricg
−1
X
α
:Txt
M→T
∗
x
t
M, i.e.v∂ →(X
−1
α
v)
ξ
◦X
−1
α
.
Formally, extremal paths for (2) can be interpreted as most probable paths
for the driving processW
twhenX α,0deﬁnes an anisotropic diﬀusion. Below, we
will identify the extremal paths as geodesics for a sub-Riemannian metric, and
we use this to ﬁnd coordinate expressions for evolutions of the paths.
3.1 Sub-Riemannian Metric on the Horizontal Distribution
We now lift the path-dependent metric deﬁned above to a sub-Riemannian metric
onHFM. For anyw,˜w∈H
(x,Xα)FM, the lift of (2)byπ ∗is the inner product
w,˜w=
ξ
X
−1
α
π∗w, X
−1
α
π∗˜w
ζ
R
n
.
The inner product induces a sub-Riemannian metricg
FM:TFM
∗
→HFM⊂
TFMby
w, g
FM(ξ)=(ξ|w),∀w∈H xFM (4)
with (ξ|w) denoting the evaluationξ(w). The metricg
FMgivesFMa (non
bracket-generating) sub-Riemannian structure [13], see also Fig.1. It is equiva-
lent to the lift
ξ∂ →π
∗
(x,X
α)
(gXα
(ξ◦π
∗
(x,X
α)
),ξ∈T
(x,Xα)FM (5)
of the metricg
Xα
above. The metric is related to the Sasaki-Mok metric onFM
[14] that extends the Sasaki metric onTM. The Sasaki-Mok metric allows paths
inFMto have derivatives in the vertical spaceVFMwhileg
FMrestricts paths
to only have derivatives inHFM. This constraint is nonholonomic thus giving
the sub-Riemannian structure.

Evolution Equations with Anisotropic Distributions and Diﬀusion PCA 7
Following [14], we let (x
i
,X
i
α
) be coordinates onFMwithX
i
α
satisfying
X
α=X
i
α
∂
∂x
i. The horizontal distribution is then spanned by thenlinearly
independent vector ﬁeldsD
j=
∂
∂x
j−Γ
hγ
j
∂
∂X
h
γwhereΓ
hγ
j
=Γ
h
ji
X
i
γ
andΓ
h
ij
are the Christoﬀel symbols for the connection∇. We denote this adapted frame
D. The vertical distribution is correspondingly spanned byD
jβ
=∂
X
j
β,and
D
h
=dx
h
,D
hγ
=Γ
hγ
j
dx
j
+dX
h
γ
constitutes a dual coframeD
∗
. The map
π
∗:HFM→TMis in coordinatesπ ∗(w
j
Dj)=w
j∂
∂x
j.
For (x, X
α)∈FM, the mapX α:R
n
→TxMis in coordinates given by the
matrix [X
i
α
] so thatX(v)=X
i
α
v
α∂
∂x
i=Xαv
α
. Withw=w
j
Djand ˜w=˜w
j
Dj,
we have
w,˜w=w
i
Di,˜w
j
Dj=
ξ
X
−1
w
i
∂
∂x
i
,X
−1
˜w
j
∂
∂x
j
ζ
=w
i
X
α
i
,˜w
j
X
α
j
R
n=δαβw
i
X
α
i
˜w
j
X
β
j
=W ijw
i
˜w
j
where [X
α
i
]istheinverseof[X
i
α
]andW ij=δαβX
α
i
X
β
j
. Deﬁne nowW
kl
=
δ
αβ
X
k
α
X
l
β
so thatW
ir
Wrj=δ
i
j
andW irW
rj
=δ
j
i
. We can then write the
sub-Riemannian metricg
FMin terms of the adapted frameD,
g
FM(ξhD
h
+ξhγ
D
hγ
)=W
ih
ξhDi, (6)
becausew, g
FM(ξ)=
Γ
w, W
jh
ξhDj
∂
=W
ijw
i
W
jh
ξh=w
i
ξi=ξhD
h
(w
j
Dj)=
ξ(w). The component matrix of the adapted frameDin the coordinates (x
i
,X
i
α
)
is
(x,Xα)LD=

I0
−ΓI

and therefore DL
(x,Xα)=

I0
ΓI

withΓ=[Γ
hγ
j
]. Similarly, for the component matrices of the dual frameD
∗
,
(x,Xα)
∗LD
∗=

IΓ
T
0I

and
D
∗L
(x,Xα)
∗=

I−Γ
T
0I

.
From (6),g
FMhasD, D
∗
components
DgFM,D
∗=

W
−1
0
00

.
Therefore,g
FMhas the following components in the coordinates (x
i
,X
i
α
):
(x,Xα)g
FM,(x,X α)
∗=
(x,Xα)LDDgFM,D
∗
D
∗L
(x,Xα)
∗=

W
−1
−W
−1
Γ
T
−ΓW
−1
ΓW
−1
Γ
T

org
ij
FM
=W
ij
,g
ijβ
FM
=−W
ih
Γ
jβ
h
,g
iαj
FM
=−Γ
iα
h
W
hj
,andg
iαjβ
FM
=Γ
iα
k
W
kh
Γ
jβ
h
.

8 S. Sommer
3.2 Geodesics forg FM
Geodesics in sub-Riemannian manifolds satisfy the Hamilton-Jacobi equations
[13]. Sinceg
FMis a lift ofg Xα
and geodesics are energy minimizing, the extremal
paths for (2) will exactly be geodesics forg
FM. In the present case, the Hamil-
tonianH(x, ξ)=
1
2
(ξ|gFM(ξ)) gives the equations
˙y
i
=g
ij
FM,y
ξj,
˙
ξi=−
1
2
∂
∂y
i
g
pq
FM,y
ξpξq.
We write (x
i
,X
i
α
) for coordinates onFMas above, and (ξ i,ξiα
) for cotangent
vectors inT
∗
FM. This gives
˙x
i
=g
ij
ξj+g
ijβ
ξjβ
=W
ij
ξj−W
ih
Γ
jβ
h
ξjβ
˙
Xi
α
=g
iαj
ξj+g
iαjβ
ξjβ
=−Γ
iα
h
W
hj
ξj+Γ
iα
k
W
kh
Γ
jβ
h
ξjβ
˙
ξ
i=−
1
2

∂
∂y
i
g
hk
y
ξhξk+
∂
∂y
i
g
hkδ
yξhξkδ
+
∂
∂y
i
g
hγk
y
ξhγ
ξk+
∂
∂y
i
g
hγkδ
yξhγ
ξkδ

˙
ξ
iα
=−
1
2

∂
∂y
iα
g
hk
y
ξhξk+
∂
∂y
iα
g
hkδ
yξhξkδ
+
∂
∂y
iα
g
hγk
y
ξhγ
ξk+
∂
∂y
iα
g
hγkδ
yξhγ
ξkδ

writingΓ
hγ
k,i
for
∂
∂y
iΓ
hγ
k
and where
∂
∂y
lg
ij
=0,
∂
∂y
lg
ijβ
=−W
ih
Γ
jβ
h,l
,
∂
∂y
lg
iαj
=
−Γ
iα
h,l
W
hj
,
∂
∂y
lg
iαjβ
=Γ
iα
k,l
W
kh
Γ
jβ
h
+Γ
iα
k
W
kh
Γ
jβ
h,l
and
∂
∂y
l
ζg
ij
=W
ij
,l
ζ
,
∂∂y
l
ζg
ijβ
=
−W
ih
,l
ζ
Γ
jβ
h
−W
ih
Γ
jβ
h,lζ
,
∂
∂y
l
ζg
iαj
=−Γ
iα
h,lζ
W
hj
−Γ
iα
h
W
hj
,l
ζ
,
∂∂y
l
ζg
iαjβ
=Γ
iα
k,lζ
W
kh
Γ
jβ
h
+Γ
iα
k
W
kh
,l
ζ
Γ
jβ
h
+Γ
iα
k
W
kh
Γ
jβ
h,lζ
withΓ
iα
h,lζ
=
∂∂y
l
ζ

Γ
i
hk
X
k
α
⊂
=δ
ζα
Γ
i
hl
and
W
ij
,l
ζ
=δ
il
X
j
ζ
+δ
jl
X
i
ζ
. Combining these expressions, we obtain the ﬂow equations
˙x
i
=W
ij
ξj−W
ih
Γ
jβ
h
ξjβ
,
˙
X
i
α
=−Γ
iα
h
W
hj
ξj+Γ
iα
k
W
kh
Γ
jβ
h
ξjβ
˙
ξ
i=W
hl
Γ
kδ
l,i
ξhξkδ
−
1
2
∀
Γ
hγ
k,i
W
kh
Γ
kδ
h
+Γ
hγ
k
W
kh
Γ
kδ
h,i

ξ
hγ
ξkδ
˙
ξ
iα
=Γ
hδ
k,iα
W
kh
Γ
kδ
h
ξhγ
ξkδ
−
∀
W
hl
,i
α
Γ
kδ
l
+W
hl
Γ
kδ
l,iα

ξ
hξkδ
−
1
2
∀
W
hk
,i
α
ξhξk+Γ
hδ
k
W
kh
,i
α
Γ
kδ
h
ξhγ
ξkδ

.
4 Cometric Formulation and Low-Rank Generator
We now investigate a cometricg
F
d
M+λgRwhereg Ris Riemannian,g
F
d
Mis a
rankdpositive semi-deﬁnite inner product arising fromdlinearly independent
tangent vectors, andλ>0. We assume thatg
F
d
Mis chosen so thatg
F
d
M+λgR
is invertible even thoughg
F
d
Mis rank-deﬁcient. The practical implication of this
construction is that a numerical implementation need not transport a fulln×n
matrix for the frame but a potentially much lower dimensionaln×dmatrix. This

Evolution Equations with Anisotropic Distributions and Diﬀusion PCA 9
situation corresponds to extracting the ﬁrstdeigenvectors in Euclidean space
PCA. When using the frame bundle to model covariances, the sum formulation
is more natural for a cometric than a metric because, with the cometric formu-
lation,g
F
d
M+λgRrepresents a sum of covariance matrices instead of a sum of
inverse covariance matrices. Thusg
F
d
M+λgRcan be intuitively thought of as
adding isotropic noise of varianceλto the covariance represented byg
F
d
M.
To pursue this, letF
d
Mdenote the bundle of rankdlinear mapsR
d
→TxM.
We deﬁne a cometric by
ξ,
˜
ξ=δ
αβ
(ξ|π
−1
∗
Xα)(
˜
ξ|π
−1
∗
Xβ)+λξ,
˜
ξ gR
forξ,
˜
ξ∈T
∗
F
d
M. The sum overα, βis forα, β=1,...,d. The ﬁrst term is
equivalent to the lift (5) of the cometricξ,
˜
ξ=
∀
ξ|g
Xα
(
ˆ
ξ)

givenX α:R
d
→
T
xM. Note that in the deﬁnition (3)ofg Xα
, the mapX αis not inverted, thus
the deﬁnition of the metric immediately carries over to the rank-deﬁcient case.
Let (x
i
,X
i
α
),α=1,...,dbe a coordinate system onF
d
M. The vertical
distribution is in this case spanned by thendvector ﬁeldsD
jβ
=∂
X
j
β. Except
for index sums being overdinstead ofnterms, the situation is thus similar
to the full-rank case. Note that (ξ|π
−1
∗
w)=(ξ|w
j
Dj)=w
i
ξi. The cometric in
coordinates is
ξ,
˜
ξ=δ
αβ
X
i
α
ξiX
j
β˜
ξ
j+λg
ij
R
ξi
˜
ξ
j=ξi
∀
δ
αβ
X
i
α
X
j
β
+λg
ij
R

˜
ξ
j=ξiW
ij˜
ξ
j
withW
ij
=δ
αβ
X
i
α
X
j
β
+λg
ij
R
. We can then write the corresponding sub-
Riemannian metricg
F
d
Min terms of the adapted frameD
g
F
d
M(ξhD
h
+ξhγ
D
hγ
)=W
ih
ξhDi (7)
because (ξ|g
F
d
M(
˜
ξ)) =
ξ
ξ,
˜
ξ
ζ
=ξ iW
ij˜
ξ
j. That is, the situation is analogous to
(6) except the termλg
ij
R
is added toW
ij
.
The geodesic system is again given by the Hamilton-Jacobi equations. As in
the full-rank case, the system is speciﬁed by the derivatives ofg
F
d
M:
∂
∂y
lg
ij
F
d
M
=
W
ij
,l
,
∂
∂y
lg
ijβ
F
d
M
=−W
ih
,l
Γ
jβ
h
−W
ih
Γ
jβ
h,l
,
∂∂y
lg
iαj
F
d
M
=−Γ
iα
h,l
W
hj
−Γ
iα
h
W
hj
,l
,
∂
∂y
lg
iαjβ
F
d
M
=Γ
iα
k,l
W
kh
Γ
jβ
h
+Γ
iα
k
W
kh
,l
Γ
jβ
h
+Γ
iα
k
W
kh
Γ
jβ
h,l
and
∂
∂y
l
ζg
ij
F
d
M
=W
ij
,l
ζ
,
∂
∂y
l
ζg
ijβ
F
d
M
=−W
ih
,l
ζ
Γ
jβ
h
−W
ih
Γ
jβ
h,lζ
,
∂∂y
l
ζg
iαj
F
d
M
=−Γ
iα
h
W
hj
,l
ζ
−Γ
iα
h,lζ
W
hj
,
∂
∂y
l
ζg
iαjβ
F
d
M
=Γ
iα
k,lζ
W
kh
Γ
jβ
h
+Γ
iα
k
W
kh
,l
ζ
Γ
jβ
h
+Γ
iα
k
W
kh
Γ
jβ
h,lζ
withΓ
iα
h,lζ
=
∂∂y
l
ζ

Γ
i
hk
X
k
α
⊂
=δ
ζα
Γ
i
hl
,W
ij
,l
=λg
R
ij
,l,andW
ij
,l
ζ
=δ
il
X
j
ζ
+δ
jl
X
i
ζ
. Note that the
introduction of the Riemannian metricg
Rimplies thatW
ij
are now dependent
on the manifold coordinatesx
i
.
5 LDDMM Landmark Equations
The cometric formulation applies immediately to the ﬁnite dimensional man-
ifolds that arise when matchingNlandmarks with the LDDMM framework

10 S. Sommer
[15]. We here use this to provide a concrete example of the ﬂow equations. The
LDDMM metric is naturally expressed as a cometric, and, using a rank-deﬁcient
inner productg
F
d
M, we can obtain a reduction of the system of equations to
2(2N+2Nd) compared to 2(2N+(2N)
2
) when the landmarks are points in
R
2
. For ease of notation, we consider only theR
2
case here. Please see [2]for
illustrations of the generated diﬀeomorphism ﬂows.
The manifoldM={(x
1
1
,x
2
1
,...,x
1
m
,x
2
m
)|(x
1
i
,x
2
i
)∈R
2
}can be represented
in coordinates by lettingi
1
,i
2
denote the ﬁrst and second indices of theith
landmark. The landmark manifold is in LDDMM given the cometricg
x(v, w)=
ﬀ
m
i,j=1
v
i
K(xi,xj)w
j
and thusg
i
k
j
l
x=K(x i,xj)
l
k
. The Christoﬀel symbols can
be written in terms of derivatives of the cometricg
ij
(recall thatδ
i
j
=g
ik
gkj=
g
jkg
ki
)[16]
Γ
k
ij
=
1
2
g
ir

g
kl
g
rs
,l
−g
sl
g
rk
,l
−g
rl
g
ks
,l
⊂
g
sj. (8)
This relation comes from the fact thatg
jm,k=−g jrg
rs
,k
gsmgives the deriv-
ative of the metric. The derivatives of the cometric is simplyg
i
k
j
l
,r
q=(δ
i
r
+
δ
j
r
)∂
x
q
rK(xi,xj)
l
k
. Using (8), derivatives of the Christoﬀel symbols can be com-
puted
Γ
k
ij ,ξ
=
1
2
g
ir,ξ
ﬀ
g
kl
g
rs
,l
−g
sl
g
rk
,l
−g
rl
g
ks
,l
ﬃ
g
sj+
1
2
g
ir
ﬀ
g
kl
g
rs
,l
−g
sl
g
rk
,l
−g
rl
g
ks
,l
ﬃ
g
sj,ξ
+
1
2
g
ir
ﬀ
g
kl
,ξ
g
rs
,l
+g
kl
g
rs
,lξ
−g
sl
,ξ
g
rk
,l
−g
sl
g
rk
,lξ
−g
rl
,ξ
g
ks
,l
−g
rl
g
ks
,lξ
ﬃ
g
sj.
This provides the full data for numerical integration of the evolution equations
onF
d
M. An implementation using the above system can be found athttp://
github.com/stefansommer/dpca.
Acknowledgement. The author wishes to thank Peter W. Michor and Sarang Joshi
for suggestions for the geometric interpretation of the sub-Riemannian metric onFM
and discussions on diﬀusion processes on manifolds. This work was supported by
the Danish Council for Independent Research and the Erwin Schr¨odinger Institute
in Vienna.
References
1. Sommer, S.: Diﬀusion Processes and PCA on Manifolds, Mathematisches
Forschungsinstitut Oberwolfach (2014)
2. Sommer, S.: Anisotropic distributions on manifolds: template estimation and most
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(eds.) IPMI 2015. LNCS, vol. 9123, pp. 193–204. Springer, Heidelberg (2015)
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dence (2002)
4. Fletcher, P., Lu, C., Pizer, S., Joshi, S.: Principal geodesic analysis for the study of
nonlinear statistics of shape. IEEE Trans. Med. Imaging23(8), 995–1005 (2004)
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6. Huckemann, S., Hotz, T., Munk, A.: Intrinsic shape analysis: geodesic PCA for
riemannian manifolds modulo isometric lie group actions. Stat. Sin.20(1), 1–100
(2010)
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opment. In: Nielsen, F., Barbaresco, F. (eds.) GSI 2013. LNCS, vol. 8085, pp.
76–83. Springer, Heidelberg (2013)
8. Tipping, M.E., Bishop, C.M.: Probabilistic principal component analysis. J. Roy.
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1178–1186 (2013)
10. Elworthy, D.: Geometric aspects of diﬀusions on manifolds. In: Hennequin, P.L.
(ed.)
´
Ecole d’
´
Et´e de Probabilit´es de Saint-Flour XV-XVII, 1985–87. Lecture Notes
in Mathematics, vol. 1362, pp. 277–425. Springer, Heidelberg (1988)
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Math. Kyoto Univ.22(1), 115–130 (1982)
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desics, and curvature. Ph.D. thesis, Brown University, Providence, USA (2008)

Barycentric Subspaces and Aﬃne
Spans in Manifolds
Xavier Pennec
(B)
Inria Sophia-Antipolis and Cˆote d’Azur University (UCA), Sophia Antipolis, France
[email protected]
Abstract.This paper addresses the generalization of Principal
Component Analysis (PCA) to Riemannian manifolds. Current meth-
ods like Principal Geodesic Analysis (PGA) and Geodesic PCA (GPCA)
minimize the distance to a “Geodesic subspace”. This allows to build
sequences of nested subspaces which are consistent with a forward com-
ponent analysis approach. However, these methods cannot be adapted to
a backward analysis and they are not symmetric in the parametrization
of the subspaces. We propose in this paper a new and more general type
of family of subspaces in manifolds:barycentric subspacesare implic-
itly deﬁned as the locus of points which are weighted means ofk+1
reference points. Depending on the generalization of the mean that we
use, we obtain the Fr´echet/Karcher barycentric subspaces (FBS/KBS)
or the aﬃne span (with exponential barycenter). This deﬁnition restores
the full symmetry between all parameters of the subspaces, contrarily to
the geodesic subspaces which intrinsically privilege one point. We show
that this deﬁnition deﬁnes locally a submanifold of dimensionkand that
it generalizes in some sense geodesic subspaces. Like PGA, barycentric
subspaces allow the construction of a forward nested sequence of sub-
spaces which contains the Fr´echet mean. However, the deﬁnition also
allows the construction of backward nested sequence which may not con-
tain the mean. As this deﬁnition relies on points and do not explicitly
refer to tangent vectors, it can be extended to non Riemannian geodesic
spaces. For instance, principal subspaces may naturally span over sev-
eral strata in stratiﬁed spaces, which is not the case with more classical
generalizations of PCA.
1 Introduction
For Principal Component Analysis (PCA) in a Euclidean space, one can equiva-
lently deﬁne the principalk-dimensional aﬃne subspace using the minimization
of the variance of the residuals (the projection of the data point to the subspace)
or the maximization of the explained variance within that aﬃne subspace. This is
due to the Pythagorean theorem, which does not hold in more general manifolds.
A second important observation is that principal components of diﬀerent orders
are nested, which allows to build forward and backward estimation methods by
iteratively adding or removing principle components.
Generalizing aﬃne subspaces to manifolds is not so obvious. For the zero-
dimensional subspace, intrinsic generalization of the mean on manifolds naturally
cﬃSpringer International Publishing Switzerland 2015
F. Nielsen and F. Barbaresco (Eds.): GSI 2015, LNCS 9389, pp. 12–21, 2015.
DOI: 10.1007/978-3-319-25040-3
2

Barycentric Subspaces and Aﬃne Spans in Manifolds 13
comes into mind: the Fr´echet mean is the set of global minima of the variance,
as deﬁned by Fr´echet in general metric spaces [5]. The set of local minima of
the variance was named Karcher mean by W.S Kendall [10] after the work of
Karcher et al. on Riemannian centers of mass ([8] see [9] for a discussion of the
naming and earlier works).
The one-dimensional component is then quite naturally a geodesic which
should passe through the mean point. Higher-order components are more diﬃcult
to deﬁne. The simplest intrinsic generalization of PCA is tangent PCA (tPCA),
which amounts to unfold the whole distribution in the tangent space at the
mean using the pullback of the Riemannian exponential map, and to compute
the principle components of the covariance matrix in the tangent space. The
method is thus based on the maximization of the explained variance. tPCA
is often used on manifolds because it is simple and eﬃcient. However, if it is
good for analyzing data which are suﬃciently centered around a central value
(unimodal or Gaussian-like data), it is often not suﬃcient for multimodal or
large support distributions (e.g. uniform on close compact subspaces).
Fletcher et al. proposed in [4] to rely on the least square distance to subspaces
which are totally geodesic at one point. These Geodesic Subspaces (GS) are
spanned by the geodesics going through one point with tangent vector restricted
to a linear subspace of the tangent space. These subspaces are only locally a man-
ifold as they are generally not smooth at the cut locus of the mean point. The pro-
cedure was coined Principle Geodesic Analysis (PGA). However, the least-square
procedure was computationally expensive, so that the authors implemented in
practice a classical tangent PCA. A real implementation of the original PGA
procedure was only provided recently by Sommer et al. [16]. PGA is intrinsic
and allows to build a sequences of embedded principal geodesic subspaces in
a forward component analysis approach by building iteratively the components
from dimension 0 (the mean point), dimension 1 (a geodesic), etc. Higher dimen-
sions are obtained iteratively by selecting the direction in the tangent space at
the mean that optimally reduce the square distance of data point to the geodesic
subspace. However, the mean always belong to geodesic subspaces even when it
is not part of the support of the distribution.
Huckemann et al. [14] proposed to start at the ﬁrst order component by
ﬁtting a geodesic to the data, not necessarily through the mean. The second
principle geodesic is chosen orthogonally to the ﬁrst one, and higher order com-
ponents are added orthogonally at the crossing point to build a geodesic sub-
space. The method was named Geodesic PCA (GPCA). Sommer [15]proposed
a method called horizontal component analysis (HCA) which uses the parallel
transport of the 2nd direction along the ﬁrst principle geodesic to deﬁne the
second coordinates, and iteratively deﬁne higher order coordinates through hor-
izontal development along the previous modes. Other principle decompositions
have been proposed, like Principle Graphs [6], extending the idea of k-means.
All the cited methods are intrinsically forward methods that build succes-
sively larger approximation spaces for the data. A notable exception is Principle
Nested Spheres (PNS), proposed by Jung, et al. [7] as a general framework
for non-geodesic decomposition of high-dimensional spheres or high-dimensional

14 X. Pennec
planar landmarks shape spaces. Subsphere or radius 0 to 1 are obtained by slic-
ing a higher dimensional sphere by an aﬃne hyperplane. The backward analysis
approach, determining a decreasing family of subspace, has been generalized to
more general manifold with the help of a nested sequence of relations [3]. How-
ever, up to know, such sequences of relationships are only known for spheres,
Euclidean spaces or quotient spaces of Lie groups by isometric actions [14].
In this paper, we keep the principle of minimizing the unexplained infor-
mation. However, we propose to replace Geodesic Subspaces by new and more
general types of family of subspaces in manifolds: Barycentric Subspaces (BS).
BS are deﬁned as the locus of points which are weighted means ofk+1 ref-
erence points. Depending on the generalization of the mean that we use on
manifolds, Fr´echet mean, Karcher mean or exponential barycenter, we obtain
the Fr´echet/Karcher barycentric subspaces (FBS/KBS) or the aﬃne span. We
show that these deﬁnition are related and locally deﬁne a submanifold of dimen-
sionk, and that they generalize in some sense the geodesic subspaces. Like PGA,
Barycentric Subspace Analysis (BSA) allows the construction of forward nested
subspaces which contains the Fr´echet mean. However, it also allows a backward
analysis which may not contain the mean. As this deﬁnition relies on points
and do not explicitly refer to tangent vectors to parametrize geodesics, a very
interesting side eﬀect is that it can also be extended to more general geodesic
spaces that are not Riemannian. For instance, in stratiﬁed spaces, it naturally
allows to have principle subspaces that span over several strata. The paper is
divided in three parts. We recall in Sect.2the background knowledge. Then, we
deﬁne in Sect.3the notions of barycentric subspaces in metric spaces and the
aﬃne spans in manifolds. Section4ﬁnally establishes important properties and
relationships between these subspaces.
2 Background Knowledge on Riemannian Manifolds
2.1 Computing in Riemannian Manifolds
We consider an embedding Riemannian manifoldMof dimensionn.The
Riemannian metric is denotedﬃ.|.ﬀ
x
on each tangent spaceT xMof the mani-
fold. The expression of the the underlying norm in a chart isδvδ
2
x
=v
T
G(x)v=
v
i
v
j
gij(x) using Einstein notations for tensor contractions. We assume the mani-
fold to be geodesically complete (no boundary nor any singular point that we can
reach in a ﬁnite time). As an important consequence, the Hopf-Rinow-De Rham
theorem states that there always exists at least one minimizing geodesic between
any two points of the manifold.
We denote by exp
x(v)theexponential mapat pointxwhich associate to each
tangent vectorv∈T
xMthe point ofMreached by the geodesic starting atx
with this tangent vector after a unit time. This map is a local diﬀeomorphism
from 0∈T
xMtoM, and we denote
−→
xy= log
x(y) its inverse: it may be deﬁned
as the smallest vector ofT
xMthat allows to shoot a geodesic fromxtoy.A
geodesic exp
x(tv) is minimizing up to a certain cut timet 0and not anymore
after. Whent
0is ﬁnite,t 0vis called a tangential cut-point and exp
x(t0v) a cut

Barycentric Subspaces and Aﬃne Spans in Manifolds 15
point. The domain of injectivityD(x)∈T xMof the exponential map can be
maximally extended up to the tangential cut-locus∂D(x)=C(x). It covers all
the manifoldMexcept thecut locusC(x) = exp
x(C(x)) which has null measure
for the Riemannian measure.
When the tangent space is provided with an orthonormal basis, the
Riemannian exponential and logarithmic maps providea normal coordinate sys-
tems atx. A set of normal coordinate systems at each point of the manifold
realize an atlas which allows to work very easily on the manifold. The imple-
mentation of exp and log maps is the basis of programming on Riemannian
manifolds, and most the geometric operations needed for statistics or image
processing can be rephrased based on them [12,13].
2.2 Taylor Expansions in Normal Coordinate Systems
We consider a normal coordinate system centered atxandx
v=exp
x(v)a
variation of the pointx. We denote byR
i
jkl
(x) the coeﬃcients of the Riemannian
curvature tensor atxand by∗a conformal gauge scale that encodes the size of
the path (in terms ofδvδ
xandδ
−→
xyδ x) normalized by the curvature. Following [2],
the Taylor expansion of the metric isg
a
b
(v)=δ
a
b
−
1
3
R
a
cbd
v
c
v
d
−
16
∇eR
a
cbd
v
e
v
c
v
d
+
O(∗
4
),and a geodesic joiningx vtoyhas initial tangent vector:
∂
log
xv
(y)
∗
a
=
−→
xy
a
−v
a
+
1
3
R
a
cbd
v
b−→
xyc−→
xyd
+
1
12
∇
cR
a
dbe
v
b−→
xyc−→
xyd−→
xye
+O(∗
4
).
Combining these two expansions, we get the expansion of the Riemannian dis-
tance:d
2
xy
(v)= dist
2
(exp
x(v),y)=δ
−→
xyδ
2
x
+(∇d
2
xy
)
T
v+
1
2
v
T
∇
2
d
2
xy
v+O(∗
3
),
where the gradient∇d
2
xy
=−2
−→
xyis−2 times the log and the Hessian is the
opposite of the diﬀerential of the log:
(∇
2
d
2
xy
)
a
b
=−[D xlog
x(y)]
a
b
=δ
a
b
−
1
3
−→
xy
c−→
xyd
R
a
cbd
−
1
12
−→
xy
c−→
xyd−→
xye
∇cR
a
dbe
+O(∗
3
).
2.3 Moments of Point Distributions
Letμ(x)=
δ
i
λiδxi
(x) be a singular distribution ofk+ 1 points onMwith
weights (λ
0,...λk) that do not sum up to zero. To deﬁne the moments of that
distribution, we have to take care that the Riemannian log and distance functions
are not smooth at the cut-locus of the points{x
i}.
Deﬁnition 1 ((k+1)-Pointed Riemannian Manifold).
Let{x
0,...xk}∈M
k+1
be a set ofk+1distinct points in the Riemannian
manifoldMandC(x
0,...xk)=∪
k
i=0
C(xi)be the union of the cut loci of these
points. We call(k+1)-pointed manifoldM
∗
(x0,...xk)=M/C(x 0,...xk)the
submanifold of the non-cut points of the points.
Since the cut locus of each point is closed and has null measure,M
∗
(x0,...xk)
is open and dense inM. Thus, it is a submanifold ofM(not necessarily con-
nected). On this submanifoldM
∗
(x0,...xk), the distance to the pointsx iand
the Riemannian log function
−→
xx
i= log
x(xi) are smooth.

16 X. Pennec
Deﬁnition 2 (Weighted Moments of a (k+1)-Pointed Manifold).
Let(λ
0,...λk)∈R
k+1
such that
δ
i
λi =0. The weightedn-order moment of a
(k+1)-pointed Riemannian manifoldM
∗
(x0,...xk)is the smooth(n,0)tensor:
M
n(x, λ)=
λ
i
λi
−→
xx
i⊗
−→
xxi...⊗
−→
xx i
σ
αη
ntimes
(1)
The 0-th order moment (the mass)M
0(λ)=
δ
i
λi=1
T
λis constant. All other
moment are homogeneous of degree 1 inλand can be normalized by dividing by
the massM
0(λ). The ﬁrst order momentM 1(x, λ)=
δ
i
λi
−→
xx
iis a smooth vector
ﬁeld on the manifoldM
∗
(x0,...xk). The second and higher order moments are
smooth (n,0) tensor ﬁelds that will be used through their contraction with the
Riemannian curvature tensor.
3 Barycentric Subspaces
In a Euclidean space, an aﬃne subspace of dimensionkis generated by a point
andknon-collinear vectors: Aﬀ(x
0,v1...vk)=

x=x 0+
δ
k
i=1
λivi,λ∈R
k

.
Alternatively, one could also generate the aﬃne span ofk+ 1 points in general
linear position using the implicit equation
δ
i
λi(xi−x) = 0 where
δ
k
i=0
λi=1.
The two deﬁnitions are equivalent whenx
i=x0+vi. The last parametrization
ofk-dimensional aﬃne submanifolds is relying on barycentric coordinates which
live in the projective spaceP
kminus the orthogonal of the line element1=(1:
1:...1):
P
∗
k
=

(λ 0:...:λ k)∈R
k+1
s.t.
λ
i
λi =0

.
Standard charts of this space are given either by the intersection of the line
elements with the “upper” unit sphereS
kofR
k+1
with north pole1/
√
k(unit
weights) or by thek-plane ofR
k+1
passing through the point1/kand orthogonal
to this vector. We call normalized weightsλ
i=λi/(
δ
k
j=0
λj) this last projection.
3.1 Fr´echet and Karcher Barycentric Subspaces in a Metric Space
The two above deﬁnitions of the aﬃne span turn out to have diﬀerent general-
izations in manifolds: the ﬁrst deﬁnition leads to geodesic subspaces, as deﬁned
in PGA and GPCA [4,14,16], while the second deﬁnition using the aﬃne span
suggests a generalization to manifolds either using the Fr´echet/Karcher weighted
mean or using an exponential barycenter.
Deﬁnition 3 (Fr´echet/Karcher Barycentric Subspaces ofk+1Points).
Let(M,dist)be a metric space and(x
0,...xk)∈M
k
bek+1distinct
reference points. The (normalized) weighted variance at pointxwith weight
λ∈P
∗
k
is:σ
2
(x, λ)=
1
2
δ
k
i=0
λ
idist
2
(x, xi)=
1
2
δ
k
i=0
λidist
2
(x, xi)/(
δ
k
j=0
λj).
The Fr´echet barycentric subspace is the locus of weighted Fr´echet means of

Barycentric Subspaces and Aﬃne Spans in Manifolds 17
these points, i.e. the set of absolute minima of the weighted variance:
FBS(x
0,...xk)={arg min x∈Mσ
2
(x, λ),λ∈P
∗
k
}. The Karcher barycentric
subspaceKBS(x
0,...xk)is deﬁned similarly with local minima instead of global
ones.
This deﬁnition restores the full symmetry of all the parameters deﬁning the sub-
spaces, contrarily to the geodesic subspaces which privilege one point. Here, we
deﬁned the notion on general metric spaces to show that it works in spaces more
general than smooth Riemannian manifolds. In a stratiﬁed space for instance,
the barycentric subspace spanned by points belonging to diﬀerent strata natu-
rally maps over all these strata. This is a signiﬁcant improvement over geodesic
subspaces used in PGA which can only be deﬁned within a regular strata.
3.2 Aﬃne Spans as Exponential Barycentric Subspaces
A second way to generalize the aﬃne span to manifolds is to see directly the
implicit barycentric coordinates equation as a weighted exponential barycenter:
Deﬁnition 4 (Aﬃne Span of a(k+1)-Pointed Riemannian Manifold).
Apointx∈M
∗
(x0,...xk)has barycentric coordinatesλ∈P
∗
k
if
M
1(x, λ)=
k
λ
i=0
λi
−→
xx
i=0. (2)
The aﬃne span of the points(x
0,...xk)∈M
k
is the set of weighted exponential
barycenters of the reference points inM
∗
(x0,...xk):
Aﬀ(x
0,...xk)={x∈M
∗
(x0,...xk)|∃λ∈P
∗
k
:M1(x, λ)=0}.
This deﬁnition is only valid onM
∗
(x0,...xk) and may hide some discontinuities
of the aﬃne span on the union of the cut locus of the reference points. Outside
this null measure set, one recognizes that Eq. (2) deﬁnes nothing else than the
critical points of the varianceσ
2
(x, λ)=
1
2
δ
i
λ
i
dist
2
(x, xi). The aﬃne span
is thus a superset of the barycentric subspaces inM
∗
(x0,...xk). However, we
notice that the variance may also have local minima on the cut-locus of the
reference points.
Let us consider ﬁeld ofn×(k+ 1) matricesZ(x)=[
−−→
xx
0,...
−−→
xx k]. We can
rewrite Eq. (2) in matrix form:M
1(x, λ)=Z(x)λ=0.Thus, we see that the
aﬃne span is controlled by the kernel of the matrix ﬁeldZ(x):
Theorem 1 (SVD Characterization of the Aﬃne Span).
LetZ(x)=U(x).S(x).V(x)
T
be a singular decomposition of the matrix ﬁelds
Z(x)=[
−−→
xx
0,...
−−→
xx k]onM
∗
(x0,...xk)(with singular values sorted in decreasing
order). The barycentric subspaceAﬀ(x
0,...xk)is the zero level-set of thek+1
singular values
k+1(x)and the subspace of valid barycentric weights is spanned
by the right singular vectors corresponding to thelvanishing singular values:
Span(v
k−l,...vk)(it is void ifl=0).

18 X. Pennec
4 Properties of Barycentric Subspaces in Manifolds
In this section, we restrict the analysis toM
∗
(x0,...xk) so that all quantities
are smooth.
4.1 Karcher Barycentric Subspaces and Aﬃne Span
InM
∗
(x0,...xk), the critical points of the weighted variance are the points of
the aﬃne span. Among these points, the local minima may be characterized by
the HessianH(x, λ)=−
δ
i
λ
i
Dxlog
x(xi) of the weighted variance. Using the
Taylor expansion of the diﬀerential of the log of Sect.2.2), we obtain:
[H(x, λ)]
a
b
=δ
a
b
−
1
3
R
a
cbd
[M2(x, λ
)]
cd
−
1
12
∇cR
a
dbe
[M3(x, λ
)]
cde
+O(∗
4
),(3)
The key factor is the contraction of the curvature with the dispersion of the
reference points: when the typically distance fromxto all the reference points
x
iis smaller than the inverse of the curvature, thenH(x, λ) is essentially close to
the identity. In the limit of null curvature, (e.g. for a Euclidean space),H(x, λ)
is simply the unit matrix. In general Riemannian manifolds, Eq. (3) only gives a
qualitative behavior. In order to obtain hard bounds on the spectrum ofH(x, λ),
one has to investigate bounds on Jacobi ﬁelds, as is done for the proof of unique-
ness of the Karcher and Fr´echet means [1,8,10,11,17]. Thanks to these proofs,
we can in fact establish that the Karcher barycentric submanifold is locally well
deﬁned around the Karcher mean.
When the Hessian is degenerated, we cannot conclude on the local minimality
without going to higher order diﬀerentials. This leads us to stratify the aﬃne
span by the index of the Hessian of the weighted variance.
Deﬁnition 5 (Regular and Positive Points ofM
∗
(x0,...xk)).
Apointx∈M
∗
(x0,...xk)is said regular (resp. positive) if the Hessian matrix
H(x, λ)is invertible (resp. positive deﬁnite) for allλin the right singular space
of the smallest singular value ofZ(x). The set of regular (resp. positive) points is
denotedReg(M
∗
(x0,...xk))(resp.Reg
+
(M
∗
(x0,...xk))). The set of positive
points of the aﬃne span is called the positive spanAﬀ
+
(x0,...xk).
Positive points are obviously regular, and in Euclidean spaces all the points
are positive and regular. However, in Riemannian manifolds, we may have non-
regular points and regular points which are non-positive.
Theorem 2 (Karcher Barycentric Subspace and Positive Span).
The positive spanAﬀ
+
(x0,...xk)is the set of regular points of the Karcher
barycentric subspaceKBS(x
0,...xk)onM
∗
(x0,...xk).
One generalization of the Fr´echet (resp. Karcher) mean is the use of the
powerαof the metric instead of the square. For instance, one deﬁnes the
median (α= 1) and the modes (α→0) as the minima of theα-variance
σ
α
(x)=
1
α
δ
k
i=0
dist
α
(x, xi). Following this idea, one could think of generalizing
barycentric subspaces to theα-Fr´echet (resp.α-Karcher) barycentric subspaces.

Barycentric Subspaces and Aﬃne Spans in Manifolds 19
In fact, it turns out that the critical points of theα-variance are just elements
of the aﬃne span with weightsλ
λ
i
=λidist
α−2
(x, xi). Thus, changing the power
of the metric just amounts to reparametrizing the barycentric weights, which
shows the notion of aﬃne span is really central.
4.2 Dimension of the Barycentric Subspace
We can locally parametrize the aﬃne span thanks to a Taylor expansion of the
constraintZ(x)λ= 0: a change of coordinatesδλinduces a change of position
δxverifyingH(x, λ)δx+Z(x)δλ=0.At the positive points, the Hessian is
invertible and the SVD characterization leads us to conclude that:
Theorem 3 (Dimension of the Barycentric Subspaces at Regular
Points).
The positive spanAﬀ
+
(x0,...,xk)(i.e. the regular KBS), is a stratiﬁed space
of dimensionkonReg(M
∗
(x0,...xk)). On them-dimensional strata,Z(x)has
exactlyk−m+1vanishing singular values.
4.3 Geodesic Subspaces as Limit of Barycentric Subspaces
By analogy with Euclidean spaces, one would expects the aﬃne span to be close
to the geodesic subspace
GS(x, w
1,...wk)=
∩
exp
x
≤
λ
k
i=1
αiwi

∈Mforα∈R
k

generated by thekindependent vectorsw
1,...wkatxwhen all the points{x i=
exp
x0
(∗wi)}1≤i≤k are converging tox 0at ﬁrst order.
In order to investigate that, we ﬁrst need to restrict the deﬁnition of the geo-
desic subspaces. Indeed, although the above classical deﬁnition is implicitly used
in most of the works using PGA, it may not deﬁne ak-dimensional submanifold
when there is a cut-locus. For instance, it is well known that geodesics of a ﬂat
square torus are either periodic or everywhere dense in a ﬂat torus submanifold
depending on whether the components of the initial velocity ﬁeld have rational
or irrational ratios. Thus, it makes sense to restrict to the part of the GS which
is limited by the cut-locus.
Deﬁnition 6 (Restricted Geodesic Submanifolds).
Letx∈Mbe a point of a Riemannian manifold andW
x={
δ
k
i=1
αiwi,α∈R
k
}
thek-dimensional linear subspace ofT
xMgenerated a k-uplet{w i}1≤i≤k ∈
(T
xM)
k
of tangent vectors atx. Recall thatD(x)⊂T xMis the maximal deﬁn-
ition domain on which the exponential map is diﬀeomorphic.
We call restricted geodesic submanifoldGS
∗
(Wx)atxgenerated by the vector
subspaceW
xthe submanifold ofMgenerated by the geodesics starting atxwith
tangent vectorsw∈W
x, but up to the ﬁrst cut-point ofxonly:
GS
∗
(Wx)=GS
∗
(x, w1,...wk)={exp
x(w),w∈W x∩D(x)}

20 X. Pennec
This restricted deﬁnition correctly deﬁnes ak-dimensional submanifold ofM,
whose completion may be a manifold with boundary.
Letx=exp
x0
(w)∈GS
∗
(Wx). Thanks to the symmetry of geodesics, we can
show that this point is solution of the barycentric equation
δ
k
i=0
λilog
x(xi)=
O(η
2
) with non-normalized homogeneous coordinatesλ i=αifor 1≤i≤k
andλ
0=η−(
δ
i
αi). These coordinates obviously sum up to zero whenη
goes to zero, which is a point at inﬁnity inP
∗
k
. In that sense, points of the
restricted geodesic submanifoldGS
∗
(W) are points at inﬁnity of the aﬃne span
Aﬀ(x, x
1,...xk) when the pointsx i=exp
x(ηwi) are converging toxat ﬁrst
order along the tangent vectorsw
i.
Theorem 4 (Restricted GS as Limit Case of the Aﬃne Span).
Points of the restricted geodesic submanifoldGS
∗
(Wx)={exp
x(w),w∈W x∩
D(x)}are points at inﬁnity inP
∗
k
of the aﬃne spanAﬀ(x, x 1,...xk)when the
pointsx
i=exp
x(ηwi)are converging toxat ﬁrst order along the tangent vectors
w
ideﬁning thek-dimensional subspaceW x⊂TxM.
5 Perspectives
We proposed in this paper three generalization of the aﬃne span ofk+ 1 points
in a manifold. These barycentric subspaces are implicitly deﬁned as the locus
of points which are weighted (Fr´echet/Karcher/exponential barycenter) means
ofk+ 1 reference points. In generic conditions, barycentric subspaces are strat-
iﬁed spaces that are locally submanifolds of dimensionk. Their singular set of
dimensionk−lcorresponds to the case wherelof the reference point belongs
to the barycentric subspace deﬁned by thek−lother reference points.
In non-generic conditions, points may coalesce along certain directions, deﬁn-
ing non local jets instead of a regulark-tuple. Geodesic subspaces, which are
deﬁned byk−1 tangent vectors at a point, do correspond (in some restricted
sense) to the limit of the aﬃne span when thek-tuple converges towards that
jet. We conjecture that this can be generalized to higher order derivatives using
techniques from sub-Riemannian geometry. This way, some non-geodesic decom-
position schemes such as loxodromes, splines and principle nested spheres could
also be seen as limit cases of barycentric subspaces.
Investigating simple manifolds like spheres and symmetric spaces will provide
useful guidelines in that direction. For instance, the closure of the barycentric
subspace ofk+ 1 diﬀerent reference points on then-dimensional sphere is the
k-dimensional great subsphere that contains the reference points. It is noticeable
that the closure of the aﬃne span generated by anyk+ 1-tuple of points of a
greatk-dimensional subsphere generate the same space, which is also a geodesic
subspace. This coincidence of spaces is due to the very high symmetry of the
sphere. For second order jets, we conjecture that we obtain subspheres of diﬀerent
radii as used in principle nested spheres (PNS) analysis.
Barycentric subspaces can be naturally nested, by deﬁning an ordering of
the reference points, which makes is suitable for a generalization of Principal
Component Analysis (PCA) to Riemannian manifolds. Several problems how-
ever remain to be investigated to use Barycentric Subspace Analysis (BSA) in

Barycentric Subspaces and Aﬃne Spans in Manifolds 21
practice. First, the optimization onk-tuple might have multiple solutions, as in
the case of spheres. Here, we need to ﬁnd a suitable quotient space similar to
the quotient deﬁnition of Grassmanians. Second, the optimization might con-
verge towards a non-local jet instead on ak-tuple, and good renormalization
techniques need to be designed to guaranty the numerical stability. Third, one
theoretically needs to deﬁne a proper criterion to be optimized by allk-tuple
fork=0...ntogether and not just a greedy approach as done by the classical
forward and backward approaches.
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geodesic analysis. Adv. Comput. Math.40(2), 283–313 (2013)
17. Yang, L.: Medians of probability measures in Riemannian manifolds and applica-
tions to radar target detection. Ph.D. thesis, Poitier University, December 2011

Dimension Reduction on Polyspheres
with Application to Skeletal Representations
Benjamin Eltzner
1(B)
, Sungkyu Jung
2
, and Stephan Huckemann
1
1
Institute for Mathematical Stochastics, University of G¨ottingen,
G¨ottingen, Germany
[email protected]
2
Department of Statistics, University of Pittsburgh, Pittsburgh, USA
Abstract.We present a novel method that adaptively deforms a poly-
sphere (a product of spheres) into a single high dimensional sphere which
then allows for principal nested spheres (PNS) analysis. Applying our
method to skeletal representations of simulated bodies as well as of data
from real human hippocampi yields promising results in view of dimen-
sion reduction. Speciﬁcally in comparison to composite PNS (CPNS), our
method of principal nested deformed spheres (PNDS) captures essential
modes of variation by lower dimensional representations.
1 Introduction
In data analysis, it is one of the big challenges to discover major modes of vari-
ation. For data in a Euclidean space this can be done by principal component
analysis (PCA) where the modes are determined by an eigendecomposition of
the covariance matrix. Notably, this is equivalent to determining a sequence of
nested aﬃne subspaces minimizing residual variance. Inspired by the eigende-
composition, [6,7] proposed PCA in the tangent space of a suitably deﬁned mean,
the notion of covariance has been generalized by [3] cf. also [2], and inspired by
minimizing residual variances, [9] proposed to ﬁnd a sequence of orthogonal best
approximating geodesics. Taking into account parallel transport, [14]proposedto
build a nested sequence of subspaces spanned by geodesics. These methods apply
to general manifolds and to some extent also to stratiﬁed spaces, e.g. to shape
spaces due to isometric (not necessarily free) actions of Lie-groups on manifolds
(cf. [8]). For spherical data, it is possible to almost entirely mimic the second
characterization of PCA by backward principal nested sphere (PNS) analysis,
proposed by [10]. Here in every step, a codimension one small hypersphere is
determined, best approximating the data orthogonally projected to the previous
small hypersphere. This method hinges on the very geometry of the sphere and
cannot be easily generalized to other spaces. For data on polyspheres (products
of spheres), which naturally occur in skeletal representations for modeling and
analysis of body organs, in composite PNS (CPNS) by [11], PNS is performed
in every factor.
In order to make PNS more directly available for polyspheres, in this com-
munication we proposeprincipal nested deformed spheres(PNDS) where we ﬁrst
cﬃSpringer International Publishing Switzerland 2015
F. Nielsen and F. Barbaresco (Eds.): GSI 2015, LNCS 9389, pp. 22–29, 2015.
DOI: 10.1007/978-3-319-25040-3
3

Dimension Reduction on Polyspheres with Application 23
deform a polysphere into a single high dimensional sphere, in a data adaptive
way, and then perform PNS on this sphere. In Sect.2we describe the proposed
deformation of the data space and in Sect.3we apply the method to simu-
lated and real data from skeletal representations and compare results to those
of CPNS. A thorough introduction and analysis of the method is deferred to a
future publication in preparation.
2 Polysphere Deformation
We assume in the following that the data space is a polysphereQ=S
di
ri
×...×S
dI
rI
and that on each individual sphere of dimensiond i∈Nand radiusr i>0the
data are conﬁned to a half sphere, 1≤i≤I. Notably, then [1] guarantees
the existence of a unique spherical meanμ
i∈S
di
ri
of the data on each individual
sphere. In the following we will deformQstepwise to a single higher-dimensional
sphereS
D
,D=d 1+...+d Iwhere the mappingP:Q−→S
D
is data-adaptive,
i.e.Pis as faithful as possible in terms of data variation.
2.1 The Construction for Unit Spheres
For equal radii, the explicit mapping is given below in (3), for varying radii the
modiﬁcation is found further down in (4) and (5).
For the following motivation, we use polar coordinates
∀1≤k≤d:x
k=
⎛
⎝
k−1
→
j=1
sinφ j
⎞
⎠cosφ
k,xd+1=
⎛
⎝
d
→
j=1
sinφ j
⎞
⎠ (1)
for the embeddingS
d
⊂R
d+1
of thed-dimensional unit sphere. We will formulate
the construction ofP=P
Irecursively, ﬁrst for two unit spheres,
P
1:S
d2
×S
d1
−→S
d2+d1
,P2:S
d3
×S
d2+d1
−→S
d3+d2+d1
, ...
where we embedS
d2
×S
d1
intoR
d2+d1+2
denoting coordinates asx 1,1,...,x1,d1+1,
x
2,1,...,x2,d2+1. Then the squared line elements of the two spheres are given by
ds
2
1
=
d1
k=1
⎛
⎝
k−1
→
j=1
sin
2
φ1,j
⎞
⎠dφ
2
1,k
,ds
2
2
=
d2
k=1
⎛
⎝
k−1
→
j=1
sin
2
φ2,j
⎞
⎠dφ
2
2,k
and the polysphere’s squared line element is given byds
2
=ds
2
2
+ds
2
1
. The line
element of the sphere, i. e. the image ofP
1is then formally deﬁned as
ds
2
=ds
2
2
+
⎛
⎝
d2→
j=1
sin
2
φ2,j
⎞
⎠ds
2
1
(2)
which can easily be checked to be a squared line element of a sphere of dimension
D=d
1+d2.

24 B. Eltzner et al.
In the next step we give a data-driven choice of coordinates and ordering of
the two unit spheres. The transformation of the line element in Eq. (2) amounts
to multiplyingds
2
1
byx
2
2,d
2+1
=1−x
2
2,1
−...−x
2
2,d
2
. This yields
⊃
x∈R
d2+d1+2

d2+1

k=1
x
2
2,k
=1=
d2
k=1
x
2
2,k
+
d1+1

k=1
(x2,d2+1x1,k)
2

Since we assumed that the data projections to each individual sphere are con-
tained in half-spheres, we may choose coordinates for each individual sphere
such thatx
i,d1+1>0 for the projections of all data points to thei-th sphere
(2≤i≤I). Often the projections are conﬁned to a half sphere centered at the
spherical meanμ
ion thei-the sphere. Then the positivex i,di+1-unit direction
can be chosen asμ
i. As the other coordinates are equally deformed, their choice
is arbitrary. Thus, the coordinates of theS
d2+d1
are given by
∀1≤k≤d
2:y j=x2,k,∀1≤k≤d 1+1 :y d2+k=x2,d2+1x1,k (3)
from which angular coordinates can be calculated by inverting the relation (1).
Usingx
2,d2+1>0 for all data, the data space is thus
S
d2+d1
=
⊃
y∈R
d2+d1+1

d2
k=1
y
2
k
+
d1+1

k=1
y
2
d
2+k=1

.
As the line element of the sphereS
d1
in Eq. (2) is multiplied by a factor≤1
for each data point, we call this sphere the “inner” sphere and note that data
variation on this sphere is reduced. High data variation on the “outer” sphere
S
d2
would lead to a greater and more uneven reduction of variation for the inner
sphere. In order to prevent this, we ﬁrst sort the spheres (i=1,...,I) such that
data variation is lowest on the last (outermost,i=I) sphere and highest on the
ﬁrst (innermost,i= 1) sphere. Indeed, if the data vary little on thei-th sphere
thenx
i,di+1is nearly one, causing little deformation.
2.2 Spheres of Diﬀerent Radii
In general, the spheres in a polysphere of interest will have diﬀerent radii. In
fact, the radius for each sphere and each datum will often be unique. Recall
that a logarithmic scale is well suited for lengths, linearizing ratios, such that
the geometric mean corresponds to the arithmetic mean on a logarithmic scale.
Hence, it is natural to deﬁne the mean radius of each individual sphere by the
geometric mean of the radii of the data points, cf. [11].
There is yet another subtlety to be dealt with. In Eq. (3), simply multiply-
ing the coordinates ofS
d2
andS
d1
by the corresponding radii, implies that all
coordinates of the sphereS
d2+d1
are in particular scaled with the radius of the
outer sphereS
d2
. This implies that the relative scaling of the spheres will only
depend on the radius of the innerS
d1
, clearly an unwanted feature. Hence, we
normalize the radii with their geometric mean

Dimension Reduction on Polyspheres with Application 25
Ri:=ri
⎛
⎝
K
→
j=1
rj
⎞
⎠
−
1
I
(i=1,...,I), rescale all coordinates of the ﬁrst unit sphere
∀1≤k≤d
1+1 :x 1,k →˜x1,k=R1x1,k, (4)
only the ﬁrstd
icoordinates of thei-th unit sphere (i=2,...I)
∀1≤k≤d
i:x i,k →˜xi,k=Rixi,k (5)
and then apply the recursive operations deﬁned in Eq. (3), using now ˜xinstead
ofx. In particular for two spheres only, we thus start with the ellipsoid
⊃
x∈R
d2+d1+1

d2
k=1
R
−2
2
x
2
2,k
+
d1+1

k=1
R
−2
1
(x2,d2+1x1,k)
2
=1

.
and only in the ﬁnal step projectyto a unit sphere. Now the ordering of the
spheres is determined by decreasing rescaled data variance where the data vari-
ance on thei-th unit sphere is rescaled by multiplication withR
i(i=1,...,I).
One of the referees pointed out that radii normalizations could also be left
variable to allow for more general optimizations. We will gratefully explore this
in further research.
3 Application to Skeletal Representations
Our method is well-suited for application to skeletal representations (s-reps), as
these contain data on a product of several spheres. An in-depth exposition of
s-reps can be found in [13], cf. also [4,5]. For the s-reps used here, we now give
a very brief review from [11].
3.1 The S-Rep Parameter Space
The basic building block of an s-rep is a two-dimensional mesh ofm×nskeletal
points which are embedded as medial as possible in the body to be described
by the s-rep so that the surface of the body splits into three parts, thenorthern
sheetabove the mesh, thesouthern sheetbelow the mesh and thecrestwhere
the two sheets meet. From each of the skeletal points emerges a spoke to a
point on the northern sheet and one to a point on the southern sheet. From
each skeletal point on the boundary of the mesh an additional spoke emerges
pointing to a point on the crest. This yields a total ofK=2mn+2m+2n−4
spokes. Figure1(b) shows an s-rep of a bent ellipsoid with 9×3 skeletal points
(yellow), northern spokes (magenta), southern spokes (blue) and crest spokes
(red). S-Reps are frequently used to model body organs in which case the spoke
directions are restricted to half spheres due to the limited ﬂexibility of organs.

26 B. Eltzner et al.
An s-rep is represented in the following product space giving the size of its
centered mesh, the lengths of the spokes, the normalized mesh-points and the
spoke directions
Q=R
+×R
K
+
×S
3mn−1
×

S
2

K
. (6)
Applying the polysphere deformation for spheres with diﬀerent radii we
obtain the data space
Q
≤
=S
5mn+2m+2n−5
. (7)
3.2 PNS, CPNS and PNDS
In PNS (cf. [10]), for data on a unit sphereS
L
a nested sequence ofl-dimensional
small-spheresM
l(l=1,...,L−1) is determined that approximates the data
best with respect to the least sum of squares of spherical residuals:
S
L
⊃M L−1⊃···⊃M 2⊃M 1⊃{μ}
At each reduction step, the residuals are recorded as signed distances from the
subsphere. In CPNS (cf. [11]), PNS is applied to every sphere occurring in the
product (7) yielding a Euclidean vector of residuals. This vector, appended by
the vector of logarithms of the sizes, is then subjected to classical PCA.
In PNDS, as proposed here, PNS is applied to the single polysphere (7)which
has been obtained by polysphere deformation for spheres with the diﬀerent radii
given by theR
+×R
K
+
factors. In particular, no further PCA step as in CPNS
is necessary.
3.3 Results
We compare the performance of PNDS to that of CPNS in terms of dimension
reduction for the following data sets.
–Hip
fullcontains s-reps ﬁtted to MRI images of 51 human hippocampi, cf. [11];
Hip
spcontains only the 66 spokes of variable length.
– Two data sets of simulated ellipsoids that have been twisted (Sim
66,1) as well
as bent and twisted (Sim
66,2)from[12] consisting of 66 unit-length spokes, cf.
Fig.1.
Overall PNDS requires fewer dimensions than CPNS to explain data varia-
tion. For the full hippocampi data, PNDS explains 90 % of the variation by 8
dimensions, CPNS by 18 dimensions (9 vs. 20 for the spokes data only). The
same eﬀect, although far less prominent is visible for the simulated data, which
is far less noisy than the real data, cf. Fig.2and Table1.
Figure3elucidates a key diﬀerence between PNDS and CPNS. The data pro-
ducing theVshape visible in components 1 and 2 of CPNS (b) obviously is spread
along several spoke spheres in (6). Because in (7) they are mapped to a single sphere,
thatVshape can be explained by a single component via PNDS (c). The residual

Dimension Reduction on Polyspheres with Application 27
Fig. 1.S-rep with 9×3 skeletal points and 66 spokes ﬁtted to a simulated bent ellipsoid,
from [12] (Color ﬁgure online).
0 10 20 30 40 50
Dimension
0
20
40
60
80
100
Variances [%]
PNDS
CPNS
(a) Variance per mode for Hip
sp
0 10 20 30 40 50
Dimension
0
20
40
60
80
100
Variances [%]
PNDS
CPNS
(b) Variance per mode for Hip
full
Fig. 2.PNDS vs. CPNS: displaying scree plots of cumulative variances for s-reps of 51
hippocampi from [11]. Right: the full data set Hip
full, left: only the spoke information
Hip
sp.
Table 1.PNDS vs. CPNS: percentage of variances explained by lower dimensional
subsets that are required to explain at least 90 % of the respective total data variance.
Sim
66,1
PNDS 92.0 92.0
CPNS 62.732.1 94.8
Sim
66,2
PNDS 88.7 7.4 96.1
CPNS 76.215.5 91.7
Hip
sp
PNDS 68.5 7.24.32.92.21.91.51.21.1 90.7
CPNS 22.810.09.16.96.15.24.83.83.02.92.52.22.11.81.61.51.41.31.11.090.9
Hip
full
PNDS 70.4 6.64.23.12.21.51.31.2 90.5
CPNS 31.2 9.68.45.14.64.24.13.53.02.32.12.01.81.31.21.21.00.9 90.5
Dimension 1 2 3 4 5 6 7 8 9 1011121314151617181920
data distances to that small circle on the two-dimensional PNDS in (c) have the
shape of a3, as visible in the ﬁrst two components of PNDS (a) and the second and
third component of CPNS (b). Higher dimensional components (already compo-
nent 3 in PNDS) only explain low variance noise as seen in (c).

28 B. Eltzner et al.
Fig. 3.PNDS vs. CPNS for simulated twisted ellipsoids (Sim 66,1): scatter plots of resid-
ual signed distances for the ﬁrst three components in (a) and (b). The data projected
to the second component (a small two-sphere) in PNDS with ﬁrst component (a small
circle) inside, is visualized in (c). As in Fig.2, subﬁgure (d) shows cumulative variances
over dimension.
4 Conclusion and Outlook
We have shown that the deformation of a polysphere data space into a single high
dimensional sphere may yield considerable enhancement in terms of dimension
reduction. For the application to skeletal representations presented here, this is
a crucial step towards a simple parametric model of body organ shapes, which
allows for better ﬁts and thus more successful automated localization of organs
in MRI images. Applications range from minimizing tissue damage in radiation
therapy or surgery to various diagnostic opportunities.

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Should a junior officer have dared to presume to have sent in his
original ideas direct to Whitehall, woe betide the day for his
immediate future and his chances for early promotion.
The above opinions are no flights of imagination; they are founded
solely on many bitter complaints which have come direct to the ears
of the writer from junior officers in both arms of the Service, whose
inventive ideas have either been summarily squashed by superior
officers, or who have been compelled in their own future interests to
stand aside, silent and disgusted, whilst they have observed others
far above them taking what credit was to be bestowed for ideas or
suggestions which were never their own, and often followed by
decoration without any patent special service.
* * * * * *
Shortly before this book went to press the author happened to meet
a naval gunner who had served for a prolonged period aboard
mystery ships. He was most enthusiastic on the subject of
camouflage, and he related how he had served in 1915 in a ship
which had one gun only, placed amidships, which was concealed by a
dummy silhouette boat.
According to his account the stunt was great. He narrated in detail
the completeness of the deception, the instantaneous manner in
which the gun was brought into action, and the success which had
attended the introduction of the idea. He affirmed that no less than
ten submarines had been sunk during the first few weeks this
invention had been first introduced. But, as he explained, one day a
vessel so fitted was attacked by two submarines at the same time,
one being on each quarter, and the secret became exposed. After
that, he added, the Germans became much more suspicious how
they approached and attacked fishing vessels, and successes fell off
considerably.
It had been an Admiralty regulation that when a submarine was sunk
and its loss proved, the successful crew was awarded £1,000 for each
submarine recorded, which was divided proportionately according to

rank. Submarines claimed to have been sunk run to over two
hundred. Many and various were the methods by which they were
sent to the bottom of the sea; but so far as a number of inventors or
the originators of ingenuity were or are concerned, it would appear
that virtue alone remains their sole reward.
* * * * * *
Since this book was accepted for press my attention has been called,
in the February number, 1920, of Pearson's Magazine, to an article by
Admiral Sims of the U.S.A. Navy, entitled "How the Mystery Ships
Fought," in which he says:
"Every submarine that was sent to the bottom, it was estimated,
amounted in 1917 to a saving of about 40,000 tons per year of
merchant shipping; that was the amount of shipping, in other
words, which the average U-boat would sink, if left unhindered
to pursue its course.
"This type of vessel (Q-boats) was a regular ship of His Majesty's
Navy, yet there was little about it that suggested warfare. Just
who invented this grimy enemy of the submarine is, like many
other devices developed by the war, unknown. It was, however,
the natural outcome of a close study of German naval methods.
The man who first had the idea well understood the peculiar
mentality of the U-boat commanders."
Extracting further paragraphs from Admiral Sims' article:
"There is hardly anything in warfare which is more vulnerable
than a submarine on the surface within a few hundred yards of a
four-inch gun. A single well-aimed shot will frequently send it to
the bottom. Indeed, a U-boat caught in such a predicament has
only one chance of escape; that is represented by the number of
seconds which it takes to get under water.
"Clearly the obvious thing for the Allies to do was to send
merchant ships armed with hidden guns along the great

highways of commerce. The crews of these ships should be
naval officers and men disguised as merchants, masters, and
sailors."
At p. 104 of the magazine Admiral Sims refers directly to my
invention as described and illustrated:
"Platforms were erected on which guns were emplaced; a
covering of tarpaulin completely hid them; yet a lever pulled by
the gun crews would cause the sides of the hatchway covers to
fall instantaneously. Other guns were placed under lifeboats,
which, by a similar mechanism, would fall apart or rise in the air
exposing the gun.
"From the greater part of 1917 from twenty to thirty of these
ships (Q-boats) sailed back and forth in the Atlantic."
The February number of the Wide World Magazine, p. 361, also
contained a most interesting article by Captain Frank H. Shaw
entitled, A "Q," and a "U," in which he describes how he personally
helped to sink a submarine with the aid of a camouflage apparatus
on the lines of my invention as illustrated:
"Meanwhile the fitters were making most of their opportunities
aboard the Penshurst (the Q-boat in question). A useful twelve-
pounder gun—one of the best bits of ordnance ever devised for
short range work—was mounted on the fore-deck. A steel ship's
lifeboat was cut in two through the keel, and so faked that on
pulling a bolt, the two halves would fall clear away. This dummy
boat was then put in place over the twelve-pounder and
effectively concealed its presence.
"So far as the outward evidence was concerned, the Penshurst
was simply carrying a spare lifeboat on deck—a not unnecessary
precaution, considering the activity of the enemy submarines."
Captain Shaw describes in stirring narrative and vivid detail how a
submarine held up his ship, how part of their crew abandoned the

ship, and how the Hun boat was lured well within easy gun-fire
range, and how my ideas worked in practice:
"The foredeck boat opened beautifully like a lily and the gun
came up, with its crew gathered round it. The twelve-pounder
was not a second behind its smaller relative. Her gunlayer, too,
was a useful man. He planted a yellow-rigged shell immediately
at the base of Fritz's conning-tower. It exploded there with
deafening report and great gouts of water flew upwards with
dark patches amongst the foam."
* * * * * *
By my friends I was disparaged for foolishness in not putting forward
a claim for compensation in connection with these ideas, followed by
an accepted invention of recognised utility. In the U.S.A. in the spring
of 1919 I heard this invention considerably lauded; in New York,
Boston, and Washington. It was also described and illustrated in
certain American periodicals.
If the figures given by Admiral Sims are true estimates, and, say, only
twenty-five submarines were sunk by the direct assistance of this
simple contrivance, then it follows that about 1,000,000 tons of
shipping were saved each year it was in active use.
Eventually I communicated with Admiral W. R. Hall, C.B., through
whom I had submitted my suggestions in the first instance. From him
I received a charming letter in which he regretted the matter had
passed beyond his department. Therefore on January 26th, 1920, I
wrote to the Secretary of the Admiralty referring by number to
previous letters conveying the secret thanks of the Lords of the
Admiralty to me in 1915 and asking him whether (now that the war
was over) I was entitled to any recognition for this invention, and if
so, how and to whom I should apply.
I wrote again on April 29th, asking for a reply to my previous letter,
but being only a civilian, I suppose he did not consider either myself

or the subject matter I enquired about worthy even of simple
acknowledgment.

CHAPTER XVIII
THE SINKING OF THE "LUSITANIA" BY
GERMAN TREACHERY
Hçï the Dastardäy Deed ïas Päanned—Cçmmemçratiçn Medaä
Prematureäy Dated—Sinking Annçunced in Beräin Befçre the Vesseä ïas
Attacked—German Jçy at the Outrage —British Secret Cçde Stçäen—
Viçäatiçns çf American Neutraäity—Faäse Messages—Authçrity fçr the
Facts.
So long as the memory of mortal man endures, this dastardly act of
German treachery will never be forgotten.
On May 7th, 1915, the SS. Lusitania, a passenger ship of 32,000 tons
of the Cunard Line, was sunk by torpedoes, fired at short range from
a German submarine off Kinsale. She carried on board 1,265
passengers and a crew of about 694 hands. From this number 1,198
were drowned, including 113 Americans and a large number of
women and children.
It is no exaggeration to say that the event staggered the humanity of
the world, yet the Kölnische Volkeszeitung on May 10th, 1915,
stated: "With joyful pride we contemplate this latest deed of our
Navy," etc. The commander of the submarine which struck the fatal
blow was decorated, and a special medal was struck in the
Fatherland commemorating the event, and dated May 5th—two days
before she was actually attacked and sunk.
A copy of it is now before the writer.
It was struck with the object of keeping alive in German hearts the
recollection of the German Navy in deliberately destroying an
unarmed passenger-ship together with 1,198 non-combatants, men,
women, and children.

On the obverse, under the legend "No Contraband" (Keine Banvare),
there is a representation of the Lusitania sinking. The designer has
put in guns and aeroplanes, which (as certified by United States
Government officials after inspection) the Lusitania did not carry, but
he has conveniently omitted to put in the women and children, which
the whole world knows she did carry.
On the reverse, under the legend "Business above all" (Geschäft über
alles), the figure of death sits at the booking-office of the Cunard
Line, and gives out tickets to passengers who refuse to attend to the
warning against submarines given by a German.
This picture seeks apparently to propound the theory that if a
murderer warns his victim of his intention, the guilt of the crime will
rest with the victim, not with the murderer.
How the foul deed was plotted and accomplished is told in concise
and simple language by Mr. John Price Jones in his book entitled,
"The German Spy in America," which has an able introduction by Mr.
Rogers B. Wood, ex-United States Assistant Attorney at New York;
also a foreword by Mr. Theodore Roosevelt.
Summarising detail and extracting bare facts from Mr. Price Jones'
work, it is shown that Germany had made her preparations long
before war was declared. She had erected a wireless station at
Sayville with thirty-five Kilowatt transmitters and had obtained special
privileges which the U.S. Government never dreamed would be so
vilely abused.
Soon after the declaration of war, Germany sent over machinery for
tripling the efficiency of the plant, via Holland, and the transmitters
were increased to a hundred Kilowatts. The whole plant was in the
hands of experts drawn from the German Navy.
On April 22nd, 1915, the German Ambassador at Washington, by
direction of Baron von Bernstorff, inserted notices by way of
advertisement warning travellers not to go in ships flying the British
flag or that of her Allies, whilst many of the ill-fated passengers

received personally private warnings; for example, Mr. A. G.
Vanderbilt had one signed "Morte."
It is also stated than one of the German spies who had helped to
conceive this diabolical scheme actually dined, the same evening the
vessel sailed, at the home of one of his American victims.
The sinking of the vessel was also published in the Berlin newspapers
before she had actually been attacked.
On reaching the edge of the war-zone, Captain Turner, who was in
charge of the Lusitania, sent out a wireless message for instructions
in accordance with his special orders.
By some means unknown the German Government had stolen a copy
of the secret code used by the British Admiralty.
A copy of this had been supplied to Sayville, which used it (inter alia)
to warn Captain Turner against submarines off the Irish coast—which
evidence was revealed at the inquest.
Sayville was very much on the alert, looking out for and expecting
Captain Turner's request for orders.
As soon as it was picked up the return answer was flashed to
"proceed to a point ten miles south of Old Head of Kinsale and run
into St. George's Channel, making Liverpool bar at midnight."
The British Admiralty also received Captain Turner's call and sent
directions "to proceed to a point seventy to eighty miles south of Old
Head of Kinsale and there meet convoy."
But the British were slow and the Germans rapid. Captain Turner
received the false message instead of the genuine one, and over a
thousand unfortunate beings were sent to their doom.
At the inquest the two messages were produced and the treachery
became apparent. Further investigations pointed direct to Sayville,
Long Island, New York, to which place the plot was traced.

The German witnesses who swore the Lusitania had guns aboard her
were indicted in America and imprisoned for perjury.
To use the wireless for any such cause as above described was
contrary to and in violation of neutrality laws; also of the United
States of America's statutes governing wireless stations.
In many chapters full of vivid detail Mr. Price Jones gives
extraordinary particulars of conspiracies and plots against persons
and property.
In scathing terms he condemns Captain Franz von Papen, von Igel
and Koenig, Captain Karl Boy-Ed, Captain Franz von Rintelen, Dr.
Heinrich F. Albert and Ambassador Dumba as spies, conspirators, or
traitors; men without conscience, whom no action, however
despicable, would stop.

CHAPTER XIX
MINISTERIAL, DIPLOMATIC, AND CONSULAR
FAILINGS
Ambassadçrs Seäected by Infäuence , nçt Merit—German Embassies
Headquarters çf Eséiçnage —Hçï Engäish Embassies Haméered Secret
Service Wçrk—Bernhardi çn the Bäçckade —Engäand's Oéen Dççrs—A
Minister 's Faiäings —British Vice-Cçnsuä's Scandaäçus Remuneratiçn—
Aäien Cçnsuäs —Hçï Itaäy ïas Brçught intç the War—Hçï the
Syméathies çf Turkey and Greece ïere Lçst—The Faiäure çf Sir Edïard
Grey—Asquith 's Prçcrastinatiçn.
The Press, it will be remembered, was during the first few years of
the war periodically almost unanimous in its outcry against the
Government, particularly the Foreign Office. Having regard to the
facts quoted, well might it be so. But the Foreign Office is somewhat
in the hands of its Ambassadors and Ministers abroad, who
unfortunately sometimes appear to put their personal dignity before
patriotism, and threaten to resign unless some ridiculous, possibly
childish, whim is not forthwith complied with. It seems hard to
believe such things can be in war-time; yet it was so. If our
Ambassadors and Ministers were selected by merit, and not by
influence, a vast improvement would at once become apparent, and
such things as were complained of would not be likely to occur or be
repeated.
One Press writer pointed out that "Great Britain lacked a watchful
policeman in Scandinavia." Perhaps he will be surprised to learn that
about the most active non-sleeping watchmen that could be found
were there soon after war started. But these watch-dogs smelt out
much too much, and most of them were caught and muzzled, or
driven away, or chained up at the instigation of the Embassies. The
heaviest chains, however, get broken, whilst the truth will ever out.

Naturally one Embassy would keep in constant touch with another,
and with regard to this question of supplying the enemy all three
Scandinavian Embassies knew, or should have known to a nicety,
precisely what was doing in each country.
We in the Secret Service had been impressively warned before
leaving England to avoid our Ambassadors abroad as we would
disciples of the devil. In so far as we possibly could we religiously
remembered and acted upon this warning. But the cruel irony of it
was, our own Ministers would not leave us alone. They seemed to
hunt us down, and as soon as one of us was located, no matter who,
or where, or how, a protest was, we were told, immediately sent to
the Foreign Office, followed by hints or threats of resignation unless
the Secret Service agent in question was instantly put out of action or
recalled to England.
I was informed that several of my predecessors had been very
unlucky in Denmark. One had been located and pushed out of the
country within a few hours of arrival. Another I heard was imprisoned
for many months. I was further very plainly told by an English official
of high degree that if the British Minister at —— became aware of my
presence and that I was in Secret Service employ, if I did not then
leave the country within a few hours of the request which would with
certainty be made, I would be handed over to the police to be dealt
with under their newly-made espionage legislation.
Considering that the German legations in Scandinavia increased their
secretaries from the two or three employed before the war to twenty
or thirty each after its outbreak; considering that it was a well-known
fact, although difficult to prove, that every German Embassy was the
local headquarters of their marvellously clever organisation of Secret
Service[15] against which our Legations possessed rarely more than
one over-worked secretary, whilst the British Embassies were a
menace rather than a help to our Secret Service, it did seem to us,
working on our own in England's cause, a cruel shame that these
men, who posed not only as Englishmen but also as directly
representing our own well-beloved King, should hound us about in a

manner which made difficult our attempts to acquire the knowledge
so important for the use of our country in its agony and dire peril.
Dog-in-the-manger-like, they persisted in putting obstacles in the way
of our doing work which they could not do themselves and probably
would not have done if they could.
If unearthing the deplorable details of the leakage of supplies to
Germany evoked disgust and burning anger in the breast of Mr. Basil
Clarke, the Special Commissioner of the Daily Mail, surely I, and
those patriotically working in conjunction with me, always at the risk
of our liberty and often at the risk of our lives, might be permitted to
feel at least a grievance against the Foreign Office for its weakness in
listening to the protests of men like these, his Britannic Majesty's
Ministers abroad; real or imaginary aristocrats appointed to exalted
positions of great dignity and possibly pushed into office by the
influence of friends at Court, or perhaps because, as the possessors
of considerable wealth, they could be expected to entertain lavishly
although their remuneration might not be excessive. Had they
remembered the patriotism and devotion to their King and country
which the immortal Horatio Nelson showed at Copenhagen a hundred
years previously, they too could just as easily have applied the
sighting glass to a blind eye, and have ignored all knowledge of the
existence of any Secret Service work or agents; unless, of course,
some unforeseen accident or circumstance had forced an official
notice upon them.
The Foreign Office would have lost none of its efficiency or its dignity,
had it hinted as much when these protests arrived; whilst England
would to-day have saved innumerable lives and vast wealth had some
of the British Ministers in the north of Europe resigned or been
removed, and level-headed, common-sensed, patriotic business men
placed in their stead as soon after war was declared with Germany as
could possibly have been arranged.
That the Germans themselves never believed England would be so
weak as to give her open doors for imports is expressed by General
Bernhardi in his "Germany and the Next War." He writes: "It is

unbelievable that England would not prevent Germany receiving
supplies through neutral countries." The following extract is from p.
157:
"It would be necessary to take further steps to secure the
importation from abroad of supplies necessary to us, since our
own communications will be completely cut off by the English.
The simplest and cheapest way would be if we obtained foreign
goods through Holland, or perhaps neutral Belgium, and could
export some part of our products through the great Dutch and
Flemish harbours.... Our own overseas commerce would remain
suspended, but such measures would prevent an absolute
stagnation of trade. It is, however, very unlikely that England
would tolerate such communications through neutral territory,
since in that way the effect of her war on our trade would be
much reduced.... That England would pay much attention to the
neutrality of weaker neighbours when such a stake was at issue
is hardly credible."
To understand what was actually permitted to happen the reader is
referred to the succeeding chapter. What possible excuse is there
which any man, that is a man, would listen to, that could be urged in
extenuation of this deplorable state of affairs and of its having been
permitted to exist and to continue so long without drastic alteration?
Our Foreign Office, hence presumably the Government, were fully
informed and knew throughout exactly what was going on. Every
Secret Service agent sent in almost weekly reports from October,
1914, onwards, emphasising the feverish activity of German agents,
who were everywhere buying up supplies of war material and food at
ridiculously high prices and transferring them to Germany with
indecent haste.
Cotton[16] and copper were particularly mentioned. Imploring
appeals were sent home by our Secret Service agents for these to be
placed on the contraband list; but no Minister explained to the nation
why, if it were feasible to make them contraband a year after the war

commenced, it was not the right thing to have done so the day after
war was declared.
German buyers openly purchased practically the whole product of the
Norwegian cod fisheries at retail prices; also the greater part of the
herring harvests. Germany absorbed every horse worth the taking,
and never before in the history of the country had so much export
trade been done, nor so much money been made by her inhabitants.
The same may be said of Sweden, with the addition that her trading
with Germany was even larger.
The British Ministers in Scandinavia seemed to carry no weight with
those with whom they were brought in contact. Their prestige had
been terribly shaken by reason of the decision to ignore entirely the
Casement affair. An Ambassador of a then powerful neutral country
referred to one of them as "what you English call a damned fool." It
was only the extraordinary ability and excellent qualities of some of
the subordinates at the Chancelleries which saved the situation.
All this had its effect in these critical times. I, who was merely a
civilian Britisher and not permanently attached to either the Army or
the Navy, and hence was not afraid to refer to a spade as a spade,
was called upon continually by others in the Service to emphasise the
true state of affairs with the Foreign Office.
Those with whom I associated in the Secret Service agreed that if the
Ministers in Scandinavia could be removed and good business men
instated at these capitals it would make a vast amount of difference
to Germany and considerably hasten along the advent of peace. But
by reason of circumstances which cannot well be revealed in these
pages, my hands were tied until such time as I could get to London
in person.
In March, 1915, I attended Whitehall, where I in no unmeasured
terms stated hard convincing facts and explained the exact position in
the north of Europe. I strongly emphasised the vital importance of
stopping the unending stream of supplies to Germany and of making

a change at the heads of the Legations mentioned. Direct access to
Sir Edward Grey was denied me, but an official of some prominence
assured me the essential facts should be conveyed to proper quarters
without delay, although the same complaints had previously been
made ad nauseam.
But facts have proved that no notice whatever of these repeated
warnings was taken, and matters went from bad to worse.
On June 21st, 1915, I had returned again to England, and I wrote
direct to Sir Edward Grey, at the Foreign Office, a letter, material
extracts from which are as follows:
"Sir,
"Being now able to speak without disobedience to orders, I am
reporting a serious matter direct to you from whom my
recommendation for Government service originates.
* * * * * *
"It is exceedingly distasteful having to speak in the semblance of
disparagement concerning anyone in His Majesty's service, and I
am only anxious to do what I believe to be right and helpful to
my country, whilst I am more than anxious to avoid any
possibility of seemingly doing the right thing in the wrong way.
But it is inconceivable that any Englishman should push forward
his false pride, or be permitted to place his personal egoism,
before his country's need; more particularly so at the present
crisis, when every atom of effort is appealed for.
"—— now being a centre and a key to so many channels through
which vast quantities of goods (as well as information) daily leak
to Germany, the head of our Legation has become a position of
vital importance. Much of the present leakage is indirectly due to
the present Minister, in whom England is indeed unfortunate.
"I therefore feel that, knowing how much depends upon even
little things, it is my bounden duty to place the plain truth clearly

before you. I have often before reported on this, so far as I
possibly could, but those whom I could report to were all so
fearful of the influences or opinions of the all-too-powerful
gentleman in question, that none of them dare utter a syllable
concerning his status or his foolish actions—although in secret
they sorrowfully admit the serious effects.
"1. Since the commencement of the war —— has committed a
series of indiscretions and mistakes, entailing a natural aftermath
of unfortunate and far-reaching consequences.
"2. Since February, 1915, he has stood discredited by the entire
—— nation, and in other parts of Scandinavia.
"3. He is bitterly opposed to the Secret Service and paralyses its
activities, although he states that his objections lie against the
department and not individuals.
* * * * * *
"In conclusion, please understand that I am in no way related to
that hopeless individual, 'the man with a grievance,' but, being
merely a civilian and having nothing whatever to expect, nor to
seek for, beyond my country's ultimate good, I can and dare
speak out; whilst the fact that in the course of my duty I went to
Kiel Harbour (despite the German compliment of a price on my
head), should be sufficient justification of my patriotism and give
some weight to my present communication.
"I have the honour to remain,
"Your obedient servant,
"Nichçäas Everitt,
"('Jim' of the B.F.S.S.)"
* * * * * *
It seems hard to believe, but this letter was passed unheeded, not
even acknowledged.

A week later, on June 28th, I wrote again, pointing out the
importance to the State of my previous communication and
emphasising further the danger of letting matters slide.
Both these letters were received at Whitehall or they would have
been returned through the Dead Letter Office. What possible reason
could there be behind the scenes that ordered and upheld such a
creed as Ruat cœlum supprimatur veritas? Or can it be ascribed to
the much-talked-of mysterious Hidden Hand?
My letters pointed only too plainly to the obvious fact that I had
information to communicate vital to the welfare of the State, which
was much too serious to commit to paper; serious information which
subservients in office dared not jeopardise their paid positions by
repeating or forwarding; information which affected the prestige of
our own King; information which might involve other countries in the
war, on one side or the other; information which it was the plain duty
of the Foreign Secretary to lose no time in making himself acquainted
with. Yet not a finger was lifted in any attempt to investigate or
follow up the grave matters which I could have unfolded, relating to
the hollowness of the Sham Blockade with its vast leakages, which
the Government had taken such pains to conceal, and to other
matters equally vital which I foreshadowed in my letter, and which
might have made enormous differences to the tide of battle and to
the welfare of nations.
No wonder the Press of all England made outcry against the Foreign
Office, as and when some of the facts relating to its dilatoriness, its
extreme leniency to all things German, and its muddle and
inefficiency in attending in time to detail gradually began to become
known.
Abroad I had heard the F.O. soundly cursed in many a Consulate and
elsewhere. I had, however, hitherto looked upon Sir Edward Grey as
a strong man in a very weak Government, a man who deserved the
gratitude of all Englishmen and of the whole Empire for great acts of
diplomacy; the man who had saved England from war more than

once; and the man who had done most to strive for peace when the
Germans insisted upon bloodshed. I would have wagered my soul
that Sir Edward Grey was the last man in England, when his country
was at war, who would have neglected his duty, or who would have
passed over without action or comment such a communication as I
had sent him.
I waited a time before I inquired. Then I heard that Sir Edward Grey
was away ill, recuperating his health salmon-fishing in N.B. But there
were others. Upon them perhaps the blame should fall.
The Foreign Office knew of, and had been fully advised, that the so-
called Blockade of Germany by our fleet was a hollow sham and a
delusion from its announced initiative. It was also fully aware that the
leakages to Germany, instead of diminishing, increased so
enormously as to create a scandal which it could hardly hope to hide
from the British public. Why, then, were these Ministers abroad
allowed to remain in office, where they had been a laughing-stock
and were apparently worse than useless? It can only be presumed
that they also had been ordered to "wait and see."
Perhaps our Ministers, particularly at the Foreign Office, believed that
they could collect, through the medium of our Consulate abroad,
practically all the information that it was necessary for our
Government to know. In peace times this might have been probable.
These self-deluded mortals seemed to have forgotten entirely that we
were at war. Furthermore, it must be admitted to our shame that our
English Consular Service in some places abroad is the poorest paid
and the least looked-after branch of Government service of almost
any nation.
Sir George Pragnell, speaking only a few days before his lamentably
sudden and untimely end, at the great meeting called by the Lord
Mayor of London at the Guildhall on January 31st, 1916, a meeting of
the representatives of Trade and Commerce from all parts of the
British Empire, said:

"Our business men maintained that our Consular Service should
consist of the best educated and the most practical business men
we could turn out. Not only should these men be paid high
salaries, but I would recommend that they should be paid a
commission or bonus on the increase of British Trade in the
places they had to look after."
If this sound, practical wisdom had only been propounded and acted
upon years ago the benefits that England would have derived
therefrom would have been incalculable. But look at the facts
regarding the countries where efficient and effective Consular Service
was most wanted during the war. In Scandinavia there were
gentlemen selected to represent us as British Vice-Consuls who
received a fixed salary of £5 per annum, in return for which they had
to provide office, clerks, telephone, and other incidentals. Although
the fees paid to them by virtue of their office and the duties they
performed may have amounted to several hundred pounds per
annum, they were compelled to hand over the whole of the fruits of
their labours to the English Government, which thus made a very
handsome profit out of its favours so bestowed. Our Foreign Office
apparently considered that the honour of the title "British Vice-
Consul" was quite a sufficient recompense for the benefits it
demanded in return, the laborious duties which it required should be
constantly attended to, and the £20 to £50 or more per annum which
their representatives were certain to find themselves out of pocket at
the end of each year. Soon after the war commenced one or two
members of the service were removed from the largest centres and
other men introduced, presumably on a special rate of pay; but in
almost all the Vice-Consulates the disgracefully mean and
unsatisfactory system above mentioned seemed to have been
continued without any attempt at reformation.
Is it to be wondered at that so many Vice-Consuls who are not
Englishmen did not feel that strong bond of sympathy either with our
Ministers abroad or with our Ministers at home, which those who

have no knowledge of the conditions of their appointment or of their
service might be led to expect existed between them?
Further light is shown upon this rotten spot in our Governmental
diplomacy management abroad by an article entitled "Scrap our Alien
Consuls," written by T. B. Donovan and published in a London paper,
February 20th, 1916, short extracts from which read as follows:
"Look up in Whitaker's Almanack for 1914 our Consuls in the
German Empire before the war—and cease to wonder that we
were not better informed. Out of a total of forty old British
Consuls more than thirty bear German names! Other nations
were not so blind.... Glance through the following astounding
list. In Sweden, twenty-four out of thirty-one British Consuls and
Vice-Consuls are non-Englishmen; in Norway, twenty-six out of
thirty; in Denmark, nineteen out of twenty-six; in Holland and its
Colonies, fourteen out of twenty-four; in Switzerland nine out of
fourteen—and several of the few Englishmen are stationed at
holiday resorts where there is no trade at all.
"And we are astonished that our blockade 'leaks at every
seam'!...
"This type of British Consul must be replaced by keen Britishers
who have the interests of their country at heart and who are at
the same time acquainted with the needs of the districts to
which they are appointed. If we could only break with red tape,
we could find numerous men, not far beyond the prime of life,
but who have retired from an active part in business, who would
gladly accept such appointments and place their knowledge at
the disposal of their fellows....
"The state of things in our Consular Service is such as no
business man would tolerate for a moment."
Turning attention to our diplomacy on the shores of the
Mediterranean and the Near East, those in the Secret Service knew
that during the early days of the war at least our Foreign Office had

nothing much to congratulate itself upon with regard to its
representatives in Italy.
For the first eight months of war an overwhelming volume of supplies
and commodities, so sought after and necessary to the Central
Powers, was permitted to be poured into and through that country
from all sources. Even the traders of the small northern neutral states
became jealous of the fortunes that were being made there. Daily
almost they might be heard saying: "Why should I not earn money by
sending goods to Germany when ten times the amount that my
country supplies is being sent through Italy?"
The tense anxiety, the long weary months of waiting for Italy to join
the Entente, are not likely to be forgotten. When at last she was
compelled to come in, it was not British cleverness in diplomacy that
caused her so to do, but the irresistible will of her own peoples, the
men in the streets and in the fields; the popular poems of Signor
D'Annunzio, which rushed the Italian Government along, against its
will, and as an overwhelming avalanche. The popular quasi-saint-like
shade of Garibaldi precipitated matters to a crisis.
"It is interesting as an object lesson in the ironies of fate to
compare the fevered enthusiasm of the Sonnino of 1881 for the
cultured Germans and Austrians, and his exuberant hatred of
France, with the cold logic of the disabused Sonnino of 1915,
who suddenly acquired widespread popularity by undoing the
work he had so laboriously helped to achieve a quarter of a
century before. European history, ever since Germany began to
obtain success in moulding it, has been full of these piquant
Penelopean Activities, some of which are fast losing their
humorous points in grim tragedy."
Thus wrote Dr. E. J. Dillon in his book of revelations, "From the Triple
to the Quadruple Alliance, or Why Italy Went to War." From cover to
cover it is full of solid, startling facts concerning the treachery and
double-dealing of the Central Powers. It shows how Italy was
flattered, cajoled and lured on to the very edge of the precipice of

ruin, disaster and disgrace; how she had been gradually hedged in,
cut off from friendly relationships with other countries, and swathed
and pinioned by the tentacles of economic plots and scheming which
rendered her tributary to and a slave of the latter-day
Conquistadores; how for over thirty years she was compelled to play
an ignominious and contemptible part as the cat's-paw of Germany;
how Prince Bulow, the most distinguished statesman in Germany, also
the most resourceful diplomatist, who by his marriage with Princess
Camporeale, and the limitless funds at his disposal, wielded
extraordinary influence with Italian senators and officials as well as at
the Vatican, dominated Italian people from the highest to the lowest;
how, in fact, the Kaiser's was the hand that for years guided Italy's
destiny. The book is a veritable mine of information of amazing
interest at the present time, given in minutest detail, authenticated
by facts, date, proof, and argument. But it is extraordinary that in this
volume of nearly 100,000 words, written by a man who perhaps, for
deep intimate knowledge of foreign politics and the histories of secret
Court intrigue, has no equal living, not a word of commendation is
devoted to the efforts made by our own British diplomacy or to the
parts played by His Britannic Majesty's Ministers and Ambassadors.
There is, however, a remote allusion on his last page but one, as
follows: "The scope for a complete and permanent betterment of
relations is great enough to attract and satisfy the highest diplomatic
ambition." This seems to be the one and only reference.
As quoted in other pages of this book, the reader will perhaps gather
that Dr. Dillon, who has been brought much in contact with the
Diplomatic Service and who has exceptional opportunities of seeing
behind the scenes, believes in the old maxims revised; for example:
De vivis nil nisi bonum.
A brief resumé of the material parts of this book which affect the
subject matter of the present one shows that on the outbreak of the
European war Italy's resolve to remain neutral provoked a campaign
of vituperation and calumny in the Turkish Press, whilst Italians in
Turkey were arrested without cause, molested by blackmailing police,
hampered in their business and even robbed of their property. But

Prince von Bulow worked hard to suppress all this and to diffuse an
atmosphere of brotherhood around Italians and Turks in Europe.
In Libya, however, Turkish machinations were not discontinued,
although they were carried on with greater secrecy. The Turks still
despatched officers, revolutionary proclamations, and Ottoman
decorations to the insurgents, and the Germans sent rifles in double-
bottomed beer-barrels via Venice. Through an accident in transit on
the railway one of these barrels was broken and the subterfuge and
treachery became revealed. The rifles were new, and most of them
bore the mark "St. Etienne," being meant not only to arm the revolt
against Italy but also to create the belief that France was
treacherously aiding and abetting the Tripolitan insurgents. And to
crown all, during the efforts of fraternisation, in German fashion,
Enver Bey's brother clandestinely joined the Senoussi, bringing
200,000 Turkish pounds and the Caliph's order "to purge the land of
those Italian traitors."
The never-to-be-forgotten "Scrap of Paper," the violation of neutral
Belgium, the shooting and burning of civilians there, the slaying of
the wounded, the torturing of the weak and helpless, at first chilled
the warm blood of humane sentiment, then sent it boiling to the
impressionable brain of the Latin race. Every new horror, every fresh
crime in the scientific barbarians' destructive progress intensified the
wrath and charged the emotional susceptibility of the Italian nation
with explosive elements. The shrieks of the countless victims of
demoniac fury awakened an echo in the hearts of plain men and
women, who instinctively felt that what was happening to-day to the
Belgians and the French might befall themselves to-morrow. The
heinous treason against the human race which materialised in the
destruction of the Lusitania completed the gradual awakening of the
Italian nation to a sense of those impalpable and imponderable
elements of the European problem which find expression in no Green
Book or Ambassadorial dispatch. It kindled a blaze of wrath and pity
and heroic enthusiasm which consumed the cobwebs of official
tradition and made short work of diplomatic fiction.

Rome at the moment was absorbed by rumours and discussions
about Germany's supreme efforts to coax Italy into an attitude of
quiescence. But these machinations were suddenly forgotten in the
fiery wrath and withering contempt which the foul misdeeds and
culmination of crimes of the scientific assassins evoked, and in pity
for the victims and their relatives.
The effect upon public sentiment and opinion in Italy, where
emotions are tensely strong, and sympathy with suffering is more
flexible and diffusive than it is even among the other Latin races, was
instantaneous. One statesman who is, or recently was, a partisan of
neutrality, remarked to Dr. Dillon that "German Kultur, as revealed
during the present war, is dissociated from every sense of duty,
obligation, chivalry, honour, and is become a potent poison, which the
remainder of humanity must endeavour by all efficacious methods to
banish from the International system. This," he went on, "is no
longer war; it is organised slaughter, perpetrated by a race suffering
from dog-madness. I tremble at the thought that our own civilised
and chivalrous people may at any moment be confronted with this
lava flood of savagery and destructiveness. Now, if ever, the
opportune moment has come for all civilised nations to join in
protest, stiffened with a unanimous threat, against the continuance
of such crimes against the human race. Europe ought surely to have
the line drawn at the poisoning of wells, the persecution of prisoners,
and the massacre of women and children."
The real cause of the transformation of Italian opinion was no mere
mechanical action; it was the inner promptings of the nation's soul.
The tide of patriotic passion was imperceptibly rising, and the cry of
completion of Italian unity was voiced in unison which culminated on
the day of the festivities arranged in commemoration of the immortal
Garibaldi. Signor D'Annunzio, the Poet Laureate of Italian Unity, was
the popular hero who set the torch to the mine of the peoples which,
when it exploded, instantly erupted parliamentary power, Ministers'
dictation, and the influences of the throne itself. It shattered the foul
system of political intrigue built up by the false Giolitti and developed

the overwhelming sentiment of an articulate nation burst into
bellicose action against the scientific barbarians; by which
spontaneous ebullition Italy took her place among the civilised and
civilising nations of Europe.
Most people who have followed events closely are convinced that
Turkey could, with judicious diplomacy, have been kept neutral
throughout the war. It was whispered in Secret Service circles that a
very few millions of money, lent or judiciously expended, would easily
have acquired her active support on the side of the Entente.
One need not probe further back in history than to the autumn of
1914 to ascertain the blundering fiasco that was made in that sphere
of our alleged activities.
Sir Edwin Pears, who has spent a lifetime in the Turkish capital and
who can hardly be designated a censorious critic, because for many
years he was the correspondent of a Liberal newspaper in London,
published, in October, 1915, a book entitled "Forty Years in
Constantinople." In that book he describes how the Turks drifted into
hostility with the Entente because the British Embassy was
completely out of touch with them. Sir Louis Mallet, H.B.M.
Ambassador, appointed in June, 1913, had never had any experience
of the country; he did not know a word of Turkish, whilst he had
under him three secretaries also ignorant of the language and of the
people. Sir Edwin Pears thus describes them:
"Mr. Beaumont, the Counsellor, especially during the days in
August before his chief returned from a visit to England, was
busy almost night and day on the shipping cases.... He also
knew nothing of Turkish, and had never had experience in
Turkey. Mr. Ovey, the First Secretary, also had never been in
Turkey, and knew nothing of Turkish. Unfortunately, also, he was
taken somewhat seriously ill. The next Secretary was Lord Gerald
Wellesley, a young man who will probably be a brilliant and
distinguished diplomatist twenty years hence, but who, like his
colleagues, had no experience in Turkey. The situation of our

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Geometric Science Of Information Second International Conference Gsi 2015 Palaiseau France October 2830 2015 Proceedings 1st Edition Frank Nielsen

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Geometric Science Of Information Second International Conference Gsi 2015 Palaiseau France October 2830 2015 Proceedings 1st Edition Frank Nielsen

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