A direct formulation for sparse PCA using semidefinite programming

TL;DR

This paper introduces a semidefinite programming approach to directly formulate sparse PCA, enabling approximation of covariance matrices with sparse eigenvectors, which is useful in various scientific fields.

Contribution

It presents a novel SDP relaxation for sparse PCA based on a modified variational eigenvalue representation, offering a new direct formulation method.

Findings

Provides a semidefinite programming relaxation for sparse PCA

Utilizes Nesterov's smooth minimization for solving the SDP

Enables approximation of covariance matrices with sparse eigenvectors

Abstract

We examine the problem of approximating, in the Frobenius-norm sense, a positive, semidefinite symmetric matrix by a rank-one matrix, with an upper bound on the cardinality of its eigenvector. The problem arises in the decomposition of a covariance matrix into sparse factors, and has wide applications ranging from biology to finance. We use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix, where cardinality is constrained, and derive a semidefinite programming based relaxation for our problem. We also discuss Nesterov's smooth minimization technique applied to the SDP arising in the direct sparse PCA method.