# Sparse Principal Component Analysis

@article{Zou2006SparsePC, title={Sparse Principal Component Analysis}, author={Hui Zou and Trevor J. Hastie and Robert Tibshirani}, journal={Journal of Computational and Graphical Statistics}, year={2006}, volume={15}, pages={265 - 286} }

Principal component analysis (PCA) is widely used in data processing and dimensionality reduction. However, PCA suffers from the fact that each principal component is a linear combination of all the original variables, thus it is often difficult to interpret the results. We introduce a new method called sparse principal component analysis (SPCA) using the lasso (elastic net) to produce modified principal components with sparse loadings. We first show that PCA can be formulated as a regression…

## 2,644 Citations

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A new PCA method is proposed to improve variable selection performance when variables are grouped, which not only selects important groups but also removes unimportant variables within identified groups to incorporate group information into model fitting.

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This work proposes a practical SPCA method in which sparse components are computed by projecting the full principal components onto a subset of the variables and shows that these components explain more than a predetermined percentage of the variance explained by the principal components.

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The method is applied on several real data examples, and diagnostic plots for detecting outliers and for selecting the degree of sparsity are provided, and an algorithm to compute the sparse and robust principal components is proposed.

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A robust sparsePCA method is proposed to handle potential outliers in the data based on the least trimmed squares PCA method which provides robust but non-sparse PC estimates and the computation time is reduced to a great extent.

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This new approach is able to find sparse PCs that are linear combinations of subsets of variables selected with respect to Type I error control and will be compared with other sparse PCA approaches by a simulation study.

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Sparse Principal Component Analysis via Joint L2,1-Norm Penalty

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This work modifications the regression model by replacing the elastic net with L 2,1-norm, which encourages row-sparsity that can get rid of the same features in different PCs, and utilizes this new "self-contained" regression model to present a new framework for graph embedding methods, which can get sparse loadings via L 1,2-norm.

Sparse Principal Component Analysis: a Least Squares approximation approach

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This work derives sparse solutions with large loadings by adding a genuine sparsity requirement to the original Principal Components Analysis objective function and proposes a Branch-and-Bound search and an iterative elimination algorithm to identify the best subset of non-zero loadings.

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