Open Access
August 2017 Computational and statistical boundaries for submatrix localization in a large noisy matrix
T. Tony Cai, Tengyuan Liang, Alexander Rakhlin
Ann. Statist. 45(4): 1403-1430 (August 2017). DOI: 10.1214/16-AOS1488


We study in this paper computational and statistical boundaries for submatrix localization. Given one observation of (one or multiple nonoverlapping) signal submatrix (of magnitude $\lambda$ and size $k_{m}\times k_{n}$) embedded in a large noise matrix (of size $m\times n$), the goal is to optimal identify the support of the signal submatrix computationally and statistically.

Two transition thresholds for the signal-to-noise ratio $\lambda/\sigma$ are established in terms of $m$, $n$, $k_{m}$ and $k_{n}$. The first threshold, $\sf SNR_{c}$, corresponds to the computational boundary. We introduce a new linear time spectral algorithm that identifies the submatrix with high probability when the signal strength is above the threshold $\sf SNR_{c}$. Below this threshold, it is shown that no polynomial time algorithm can succeed in identifying the submatrix, under the hidden clique hypothesis. The second threshold, $\sf SNR_{s}$, captures the statistical boundary, below which no method can succeed in localization with probability going to one in the minimax sense. The exhaustive search method successfully finds the submatrix above this threshold. In marked contrast to submatrix detection and sparse PCA, the results show an interesting phenomenon that $\sf SNR_{c}$ is always significantly larger than $\sf SNR_{s}$ under the sub-Gaussian error model, which implies an essential gap between statistical optimality and computational efficiency for submatrix localization.


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T. Tony Cai. Tengyuan Liang. Alexander Rakhlin. "Computational and statistical boundaries for submatrix localization in a large noisy matrix." Ann. Statist. 45 (4) 1403 - 1430, August 2017.


Received: 1 October 2015; Revised: 1 April 2016; Published: August 2017
First available in Project Euclid: 28 June 2017

zbMATH: 06773278
MathSciNet: MR3670183
Digital Object Identifier: 10.1214/16-AOS1488

Primary: 62C20
Secondary: 90C27

Keywords: Computational boundary , computational complexity , Detection , lower bounds , minimax , planted clique , signal-to-noise ratio , statistical boundary , submatrix localization

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 4 • August 2017
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