Kernel PCA
Implement Kernel PCA using the RBF kernel.
Standard PCA finds linear projections. Kernel PCA applies the kernel trick to find nonlinear projections:
- Compute the RBF kernel matrix: Kij=exp(−∥xi−xj∥2/(2σ2))
- Center the kernel matrix: K~=K−1nK−K1n+1nK1n where 1n is the n×n matrix with all entries 1/n
- Compute eigenvalues/eigenvectors of K~
- Return the top k eigenvector columns scaled by λ
For simplicity, this problem uses n=3 points and k=1. Return the projected values rounded to 4 decimal places.
Note: Use power iteration to find the dominant eigenvector.
Example:
X = [[0], [1], [2]] sigma = 1.0 k = 1
[[-0.6575], [0.0], [0.6575]]
- The RBF kernel matrix K is computed using the formula Kij=exp(−∥xi−xj∥2/(2σ2)), resulting in a 3×3 matrix.
- The kernel matrix is then centered to obtain K~=K−1nK−K1n+1nK1n, where 1n is a 3×3 matrix with all entries 1/3.
- Using power iteration, the dominant eigenvector of K~ is found, and its components are scaled by the square root of the corresponding eigenvalue λ.
- The resulting vector is then rounded to 4 decimal places, yielding the projected values [−0.6575,0.0,0.6575].
Constraints:
- X: 2D list of data points (n x d)
- sigma: RBF kernel width
- k: number of components (1)
- Return 2D list (n x k) of projected values, rounded to 4 decimal places
Background Knowledge
Kernel PCA is a technique used for dimensionality reduction that extends traditional Principal Component Analysis (PCA) by using the kernel trick to find nonlinear projections. In traditional PCA, we find linear projections that capture the most variance in the data. However, many real-world datasets have nonlinear relationships between features, which cannot be captured by linear projections. This is where Kernel PCA comes in – it uses a kernel function to map the data into a higher-dimensional space, where linear projections can capture nonlinear relationships in the original space.
The RBF (Radial Basis Function) kernel is a commonly used kernel function, which computes the similarity between two points xi and xj as Kij=exp(−∥xi−xj∥2/(2σ2)). The kernel matrix K is a matrix where the entry at row i and column j is the similarity between points xi and xj. The centering step is necessary to ensure that the kernel matrix is zero-mean, which is a requirement for PCA. Centering involves subtracting the mean of each row and column from the kernel matrix.
The eigendecomposition of the centered kernel matrix K~ gives us the eigenvalues and eigenvectors, which represent the amount of variance captured by each projection and the direction of the projection, respectively. The power iteration method is an efficient way to find the dominant eigenvector, which corresponds to the projection that captures the most variance. By scaling the eigenvector columns by λ, we can obtain the projected values.
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