Algorithm

Center data. Compute covariance matrix. Eigendecompose. Top-k eigenvectors = principal components. Project.

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SVD version

X = UΣV^T. Principal components = right singular vectors V. Avoids explicit covariance computation.

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Variance explained

Fraction of variance retained = sum of top-k eigenvalues / sum of all. Choose k for 95% typically.

Complexity

Covariance: O(N·d²). Eigendecomp: O(d³). Randomized SVD: O(N·d·k) for k components.