## The Annals of Statistics

### Computational barriers in minimax submatrix detection

#### Abstract

This 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 $p\to\infty$, if the submatrix size $k=\Theta(p^{\alpha})$ for any $\alpha\in(0,{2}/{3})$, 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 $k=\Theta(p^{\alpha})$ for any $\alpha\in({2}/{3},1)$, 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.

#### Article information

Source
Ann. Statist., Volume 43, Number 3 (2015), 1089-1116.

Dates
Revised: December 2014
First available in Project Euclid: 15 May 2015

https://projecteuclid.org/euclid.aos/1431695639

Digital Object Identifier
doi:10.1214/14-AOS1300

Mathematical Reviews number (MathSciNet)
MR3346698

Zentralblatt MATH identifier
1328.62354

Subjects
Primary: 62H15: Hypothesis testing
Secondary: 62C20: Minimax procedures

#### Citation

Ma, Zongming; Wu, Yihong. Computational barriers in minimax submatrix detection. Ann. Statist. 43 (2015), no. 3, 1089--1116. doi:10.1214/14-AOS1300. https://projecteuclid.org/euclid.aos/1431695639

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