Statistics Papers

Document Type

Journal Article

Date of this Version

2015

Publication Source

The Annals of Statistics

Volume

43

Issue

3

Start Page

1089

Last Page

1116

DOI

10.1214/14-AOS1300

Abstract

his paper studies the minimax detection of a small submatrix of elevated mean in a large matrix contaminated by additive Gaussian noise. To investigate the tradeoff between statistical performance and computational cost from a complexity-theoretic perspective, we consider a sequence of discretized models which are asymptotically equivalent to the Gaussian model. Under the hypothesis that the planted clique detection problem cannot be solved in randomized polynomial time when the clique size is of smaller order than the square root of the graph size, the following phase transition phenomenon is established: when the size of the large matrix p → ∞, if the submatrix size k = Θ(pα) for any α ∈ (0,2/3), computational complexity constraints can incur a severe penalty on the statistical performance in the sense that any randomized polynomial-time test is minimax suboptimal by a polynomial factor in p; if k = Θ(pα) for any α ∈ (2/3,1), minimax optimal detection can be attained within constant factors in linear time. Using Schatten norm loss as a representative example, we show that the hardness of attaining the minimax estimation rate can crucially depend on the loss function. Implications on the hardness of support recovery are also obtained.

Copyright/Permission Statement

The original and published work is available at: https://projecteuclid.org/euclid.aos/1431695639#abstract

Keywords

Asymptotic equivalence, high-dimensional statistics, computational complexity, minimax rate, planted clique, submatrix detection

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Date Posted: 27 November 2017

This document has been peer reviewed.