PCA via SVD — the connection.
mediumAnswer
- For centered X (n × p), SVD gives X = UΣV'.
- Principal directions = columns of V (right singular vectors).
- Principal scores = UΣ.
- Explained variance ratios ∝ .
- More numerically stable than eigendecomposing X'X (which squares condition number).
- Standard scikit-learn PCA uses SVD internally.
- Randomized SVD (Halko-Tropp) approximates top-k in O(np log k) — big-data default.
Check yourself — multiple choice
- Random
- SVD X = UΣV' → V = principal directions, UΣ = scores; more stable than eigen(X'X); randomized SVD for scale
- Same as t-SNE
- Only for small X
PCA via SVD: V = directions, UΣ = scores; randomized SVD at scale.
#dimensionality-reduction#pca
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