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Comparing PLS vs SVD in Dimensionality Reduction

Explore the differences between Partial Least Squares (PLS) and Singular Value Decomposition (SVD) in dimensionality reduction. Learn about the intuitions behind both techniques, with PLS focusing on covariance maximization and SVD on variance maximization. Discover how PLS addresses the limitations of linear regression and allows for partial decomposition of datasets. Dive into the practical applications of PLS and SVD in datasets with sparse, high-dimensional data. Understand the impact of overfitting in PLS and how it compares to SVD in various dimensions.

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Comparing PLS vs SVD in Dimensionality Reduction

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  1. PLS Vs. SVD inDimensionalityReduction Paul Hsiung December 3, 2002 16-811

  2. Problem • Curse of dimensionality • Very sparse data: • A lot of 0’s • Some attributes irrelevant • Others are repeated • Many machine learning algorithm are infeasible at high dimensions. • Compound dataset example…

  3. SVD Quick Review • Find the axis with greatest variance. • Project your data unto this axis. • Let the top n eigenvectors be the space of your new decomposed data. x2 e2 e1 x1

  4. Partial Least Squares: Intuition 1 • SVD max the variance of X, PLS max the covariance of X and Y. • SVD does not factor in Y when decomposing. • A good picture would be…

  5. Linear Regression • Given data output Y, input X • Find a w such that wTx best approximates Y in the least square sense. • The magical formula for w is y  w   1  x

  6. PLS: Intuition 2 • Problem with linear regression is… • PLS does as the name says, it finds the least squares except it’s partial. • As it builds Bpls, it will decompose X into T • We can control how many dimension T has by the number of iterations in PLS

  7. PLS: No Guts, No Glory

  8. PLS: Aftermath • Collect all small t1…tn into T. Same for P, B, and W. • Notice that T is s x n and that’s our decomposed dataset. • We define . R will transform any X into T. • Prediction is done byQ is a column of 1’s.

  9. Dataset • Training set is 26,000 by 6000. Test is 1,400 by 6000. • Single output • Very sparse… lots of 0’s • Used ROC curve to rank results.

  10. PLS: Overfitting

  11. Conclusion • PLS at dim 10 is equivalent to SVD at dim 100. But SVD is slightly better in the high dimensions. • PLS tends to overfit after dim 10. • PLS as a predictor works pretty well.

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