Sparse PCA with Oracle Property

TL;DR

This paper introduces new estimators for sparse PCA that can exactly recover the true support and achieve optimal convergence rates in high-dimensional settings without relying on strong model assumptions.

Contribution

It proposes a family of semidefinite relaxation-based estimators that recover the support and improve convergence rates for sparse PCA, surpassing existing methods.

Findings

One estimator recovers the true support with high probability.

Another estimator achieves a sharper convergence rate.

The methods do not depend on the spiked covariance model.

Abstract

In this paper, we study the estimation of the $k$-dimensional sparse principal subspace of covariance matrix $\Sigma$ in the high-dimensional setting. We aim to recover the oracle principal subspace solution, i.e., the principal subspace estimator obtained assuming the true support is known a priori. To this end, we propose a family of estimators based on the semidefinite relaxation of sparse PCA with novel regularizations. In particular, under a weak assumption on the magnitude of the population projection matrix, one estimator within this family exactly recovers the true support with high probability, has exact rank-$k$, and attains a $\sqrt{s/n}$ statistical rate of convergence with $s$ being the subspace sparsity level and $n$ the sample size. Compared to existing support recovery results for sparse PCA, our approach does not hinge on the spiked covariance model or the limited correlation condition. As a complement to the first estimator that enjoys the oracle property, we prove that, another estimator within the family achieves a sharper statistical rate of convergence than the standard semidefinite relaxation of sparse PCA, even when the previous assumption on the magnitude of the projection matrix is violated. We validate the theoretical results by numerical experiments on synthetic datasets.