Sparse Principal Component of a Rank-deficient Matrix

TL;DR

This paper presents a polynomial-time algorithm for finding the optimal sparse principal component of a rank-deficient matrix, leveraging auxiliary variables and a polynomially bounded set of candidate index-sets.

Contribution

It introduces a novel approach using auxiliary spherical variables and polynomially bounded candidate sets to efficiently compute the sparse principal component.

Findings

Polynomial-time algorithm for sparse PCA in rank-deficient matrices

Existence of a polynomially bounded set of candidate index-sets

Optimal sparse principal component can be identified efficiently

Abstract

We consider the problem of identifying the sparse principal component of a rank-deficient matrix. We introduce auxiliary spherical variables and prove that there exists a set of candidate index-sets (that is, sets of indices to the nonzero elements of the vector argument) whose size is polynomially bounded, in terms of rank, and contains the optimal index-set, i.e. the index-set of the nonzero elements of the optimal solution. Finally, we develop an algorithm that computes the optimal sparse principal component in polynomial time for any sparsity degree.