[Paper Review] Sparse PCA from Sparse Linear Regression

Guy Bresler, Sung Min Park+1 more|arXiv (Cornell University)|Nov 25, 2018

Blind Source Separation Techniques3 citations

TL;DR

This paper presents a novel reduction that transforms any black-box sparse linear regression (SLR) solver into a sparse principal component analysis (SPCA) algorithm. By regressing each variable on the others using SLR, the method achieves near-optimal statistical guarantees for both hypothesis testing and support recovery in the single spiked covariance model, matching state-of-the-art performance with provable error bounds.

ABSTRACT

Sparse Principal Component Analysis (SPCA) and Sparse Linear Regression (SLR) have a wide range of applications and have attracted a tremendous amount of attention in the last two decades as canonical examples of statistical problems in high dimension. A variety of algorithms have been proposed for both SPCA and SLR, but an explicit connection between the two had not been made. We show how to efficiently transform a black-box solver for SLR into an algorithm for SPCA: assuming the SLR solver satisfies prediction error guarantees achieved by existing efficient algorithms such as those based on the Lasso, the SPCA algorithm derived from it achieves near state of the art guarantees for testing and for support recovery for the single spiked covariance model as obtained by the current best polynomialtime algorithms. Our reduction not only highlights the inherent similarity between the two problems, but also, from a practical standpoint, allows one to obtain a collection of algorithms for SPCA directly from known algorithms for SLR. We provide experimental results on simulated data comparing our proposed framework to other algorithms for SPCA.

Motivation & Objective

• To establish a formal algorithmic connection between sparse linear regression (SLR) and sparse principal component analysis (SPCA), two problems with similar statistical and computational trade-offs.
• To develop a general, efficient framework that converts any SLR solver into a provably effective SPCA algorithm with theoretical guarantees.
• To achieve near-optimal performance in both hypothesis testing and support recovery under the single spiked covariance model.
• To demonstrate robustness of the framework to data rescaling, such as using correlation matrices instead of covariance matrices.

Proposed method

• The method regresses each coordinate of the data on all other coordinates using a black-box SLR solver, treating each variable as a response and the rest as predictors.
• For each coordinate $ i $, the method computes a test statistic $ Q_i $ based on the prediction error from the SLR fit.
• A hypothesis test is performed by checking if any $ Q_i $ exceeds a threshold proportional to $ \frac{k \log d}{n} $, distinguishing isotropic from spiked covariance models.
• For support recovery, the method identifies the support of the spike by selecting coordinates where $ Q_i $ exceeds the threshold, under a minimum signal size assumption.
• The framework is robust to data rescaling, as shown both theoretically and empirically, making it effective even when diagonal thresholding fails.
• Theoretical analysis relies on concentration inequalities and union bounds to control type I and type II error probabilities under high-dimensional asymptotics.

Experimental results

Research questions

• RQ1 Can a black-box SLR solver be systematically transformed into an SPCA algorithm with provable statistical guarantees?
• RQ2 Does the proposed reduction achieve near-optimal signal strength thresholds for hypothesis testing and support recovery in the single spiked covariance model?
• RQ3 How does the framework perform under data rescaling, such as using correlation matrices instead of covariance matrices?
• RQ4 Is the reduction robust to variations in the signal structure, such as unequal non-zero entries in the spike vector?
• RQ5 How does the method compare empirically to existing SPCA algorithms like covariance thresholding and truncated power methods?

Key findings

• The proposed SPCA algorithm achieves a hypothesis testing threshold of $ \theta \gtrsim \sqrt{\frac{k^2 \log d}{n}} $, matching the phase transition of diagonal thresholding and minimal dual perturbation up to constant factors.
• For support recovery, the algorithm succeeds with high probability when $ \theta \gtrsim \sqrt{\frac{k^2 \log d}{n}} $, assuming each non-zero entry in the spike vector is at least $ \Omega(1/\sqrt{k}) $.
• The framework outperforms covariance thresholding in experiments, particularly when data is rescaled, due to its robustness to correlation-based preprocessing.
• The method is robust to rescaling, such as using a Pearson correlation matrix, which renders diagonal thresholding ineffective but does not degrade performance.
• The runtime is $ \tilde{O}(nd^2) $, dominated by the cost of computing the sample covariance matrix, and is competitive with spectral and SDP-based methods in theory.
• Empirical results show that using standard SLR solvers (e.g., thresholded Lasso) as the black-box yields strong performance, validating the practicality of the framework.