Statistical and computational trade-offs in estimation of sparse principal components

Abstract

In recent years, sparse principal component analysis has emerged as an extremely popular dimension reduction technique for highdimensional data. The theoretical challenge, in the simplest case, is to estimate the leading eigenvector of a population covariance matrix under the assumption that this eigenvector is sparse. An impressive range of estimators have been proposed; some of these are fast to compute, while others are known to achieve the minimax optimal rate over certain Gaussian or sub-Gaussian classes. In this paper, we show that, under a widely-believed assumption from computational complexity theory, there is a fundamental trade-off between statistical and computational performance in this problem. More precisely, working with new, larger classes satisfying a restricted covariance concentration condition, we show that there is an effective sample size regime in which no randomised polynomial time algorithm can achieve the minimax optimal rate. We also study the theoretical performance of a (polynomial time) variant of the well-known semidefinite relaxation estimator, revealing a subtle interplay between statistical and computational efficiency.Tribute: Peter was a remarkable person: not only a prolific and highly influential researcher, but also someone with a wonderful warmth and generosity of spirit. He was a great inspiration to so many statisticians around the world. We are deeply saddened that he is no longer with us, and dedicate this paper to his memory. Further personal reflections on Peter Hall's life and work from the third author can be found in Samworth (2016).

High Dimensional Change Point Estimation via Sparse Projection

“…() in the context of the lasso in high dimensional linear models, or Johnstone and Lu (), or Wang et al . () in the context of sparse principal component analysis.…”

Sparse Principal Component Analysis via Axis-Aligned Random Projections

“…(), Ma () and Wang et al . ()). To give a flavour of these results, let denote the set of all estimators of v 1 , i.e.…”

“…As an alternative method, in Figs (a)–(d), we also present the corresponding results for the variants of Wang et al . () of the semidefinite programming algorithm that was introduced by d’Aspremont et al . ().…”

A Literature Survey on High-Dimensional Sparse Principal Component Analysis

“…The main questions needed to be answered in sparse PCA is whether there has an algorithm not only asymptotically consistent but also computationally efficient. Theoretical research from statistical guarantees view of sparse PCA includes consistency [2,8,14,38,41,50,53,55], minimax risk bounds for estimating eigenvectors [40,[42][43]45,61], optimal sparsity level detection [4,44,48,59] and principal subspaces estimation [5,[15][16]36,9,40,51,57] have been established under various statistical models. Because most of the methods based on spiked covariance model, so we firstly given an introduction about spiked variance model and then give a high dimensional sparse PCA theoretical analysis review from above several aspects.…”

“…They established minimax rates for estimation under 2 l loss with q l -penalized estimators with suitably model parameters. Wang et al [59] considered the question of whether it is possible to find an estimator of 1 u that is computable in polynomial time, and it attained the minimax optimal rate of convergence 1 u . They showed that no randomized polynomial time algorithm can achieve the minimax optimal rate.…”