Physical Significance of Eigenvalues in ML Sophia Sarah2023UCA1863.pptx

eigenvalues and eigenvectors

1. Physical Significance of Eigenvalues in Machine Learning Insights, Applications, and Practical Use Cases SOPHIA SARAH 2023 UCA1863

2. What Are Eigenvalues and Eigenvectors? Eigenvalue ( λ) : A scalar indicating how much an eigenvector is scaled during a linear transformation. Eigenvector (v) : A vector that remains in the same direction under a transformation. Mathematical Equation : Av= λ v Key Idea : Eigenvalues and eigenvectors reveal the structure and behavior of data when represented as a matrix.

3. Dimensionality Reduction with PCA Principal Component Analysis (PCA): Identifies the directions (principal components) of maximum variance in data. Eigenvalues represent the amount of variance captured by each principal component. Example: Data with 10 features → PCA reduces it to 2-3 key features. Eigenvalues show which features hold most of the variance. Visual: Scree plot showing eigenvalues for components. Scatterplot of PCA-reduced data.

4. Feature Importance in Machine Learning How Transformations Use Eigenvalues : Eigenvalues determine how much the data is stretched or compressed. Example : A covariance matrix scaling transforms data along its principal directions. Importance : Helps understand data behavior under linear transformations like rotations or projections. Visual : Example of a geometric transformation before and after.

5. Stability in Optimization Hessian Matrix in Optimization : The eigenvalues of the Hessian matrix (second derivative of loss function) determine stability: All eigenvalues > 0 → Local minimum. All eigenvalues < 0 → Local maximum. Mixed eigenvalues → Saddle point. Applications : Neural network training (gradient descent). Convex optimization problems. Visual : Loss surface with annotations for eigenvalue roles.

6. Spectral Clustering and Graph Analysis Spectral Clustering : Uses eigenvalues of the graph Laplacian to partition a graph into clusters. Physical Significance : Eigenvalues determine connectivity strength within the graph. Smaller eigenvalues relate to more connected components. Applications : Image segmentation, community detection in networks. Visual : Graph with clustering and eigenvalue annotations.

7. Understanding Data Correlations Covariance Matrix and Eigenvalues : The covariance matrix summarizes correlations between features. Eigenvalues indicate the strength of these correlations. Applications : Helps detect multicollinearity and redundant features. Example : Eigenvalues close to zero imply weak correlations (less importance). Visual : Heatmap of covariance matrix with eigenvalue analysis. .

8. Applications in Machine Learning : 1. Dimensionality Reduction (PCA) : Reduce high-dimensional data while preserving variance. Feature Selection : Identify and retain important features. Optimization Stability : Analyze convergence using Hessian eigenvalues. Clustering (Spectral Clustering) : Partition graphs effectively. Data Transformation : Understand how transformations scale data. Data Correlation : Identify relationships and redundancies in features. Takeaway : Eigenvalues bridge mathematical theory with practical problem-solving in machine learning.

9. Python Implementation: Physical Significance of Eigenvalues and Eigenvectors in Machine Learning

10. Python Implementation: Physical Significance of Eigenvalues and Eigenvectors in Machine Learning

11. THANK YOU