Kernel PCA with RBF Kernel
Dimensionality Reduction & Anomaly Detection DS practice problem on Onlearn.
Difficulty: medium.
Topics: Kernel PCA with RBF Kernel, Radial Basis Function, Gram Matrix, Centering in Feature Space, Kernel Trick, Spectral Embedding, Linear Algebra, Statistical Learning Theory, Optimization Theory, Manifold Learning, Information Geometry, Kernel Methods, Eigenvalue Decomposition, Non-linear Dimensionality Reduction, Reproducing Kernel Hilbert Spaces, Distance Metrics.
Implement a function kernel pca rbf that performs Kernel Principal Component Analysis using the Radial Basis Function (RBF) kernel. Kernel PCA is a nonlinear dimensionality reduction technique that uses the kernel trick to project data into a higher dimensional feature space where linear PCA is performed. Given an input data matrix X of shape (n samples, n features), the number of components to extract, and the gamma parameter for the RBF kernel, your function should: 1. Compute the RBF kernel matrix 2. Center the kernel matrix appropriately 3. Perform eigendecomposition 4. Return the transformed data with the specified number of components The RBF kernel between two points x and y is defined as K(x, y) = exp( gamma ||x y||^2). Note: To ensure consistent results, enforce a sign convention where the element with the largest absolute value in each eigenvector column is positive. Sort principal components by eigenvalue in descending order.