Kernel PCA

PCA in feature space

Plain PCA finds straight directions, so it cannot unfold curved structure. Kernel PCA runs PCA on a kernel matrix instead, which implicitly maps the data into a richer feature space.

Implement kernel_pca(X, n_components, gamma=None). Return an array of shape (n, n_components).

1. Build the kernel matrix K. If gamma is None, use the linear kernel K = X Xᵀ. Otherwise use the RBF kernel K(i, j) = exp(−gamma · ||x_i − x_j||²).
2. Center the kernel: Kc = K − 1ₙK − K1ₙ + 1ₙK1ₙ, where 1ₙ is the n × n matrix with entries 1/n.
3. Take the eigendecomposition of Kc. Keep the n_components largest eigenvalues with their eigenvectors.
4. Scores are the eigenvectors scaled by the square roots of their eigenvalues. Eigenvalues that are not positive contribute zero columns.

Constraints

• 1 ≤ n_components ≤ n, otherwise raise ValueError.
• gamma must be positive when given, otherwise raise ValueError.
• Do not modify X.

Hints