Computational Lower Bounds for Sparse PCA

TL;DR

This paper establishes computational lower bounds for sparse PCA detection, showing a fundamental trade-off between statistical power and computational efficiency under certain complexity assumptions.

Contribution

It introduces a semidefinite programming based test for sparse PCA and proves its optimality among efficient methods under planted clique conjecture assumptions.

Findings

The proposed SDP-based test achieves near-optimal detection thresholds.

Computational lower bounds match the performance of the proposed method.

Results are conditioned on the hardness of the planted clique problem.

Abstract

In the context of sparse principal component detection, we bring evidence towards the existence of a statistical price to pay for computational efficiency. We measure the performance of a test by the smallest signal strength that it can detect and we propose a computationally efficient method based on semidefinite programming. We also prove that the statistical performance of this test cannot be strictly improved by any computationally efficient method. Our results can be viewed as complexity theoretic lower bounds conditionally on the assumptions that some instances of the planted clique problem cannot be solved in randomized polynomial time.