Computational and Statistical Boundaries for Submatrix Localization in a Large Noisy Matrix

TL;DR

This paper investigates the limits of efficiently detecting a signal submatrix in a noisy large matrix, revealing a gap between what is statistically possible and what is computationally feasible.

Contribution

It establishes computational and statistical thresholds for submatrix localization and introduces efficient algorithms that operate near the computational boundary.

Findings

Identifies a computational threshold  SNR_c for polynomial-time success.

Defines a statistical threshold  SNR_s below which detection is impossible.

Shows a significant gap between computational and statistical boundaries.

Abstract

The interplay between computational efficiency and statistical accuracy in high-dimensional inference has drawn increasing attention in the literature. In this paper, we study computational and statistical boundaries for submatrix localization. Given one observation of (one or multiple non-overlapping) signal submatrix (of magnitude $\lambda$ and size $k_m \times k_n$) contaminated with a noise matrix (of size $m \times n$), we establish two transition thresholds for the signal to noise $\lambda/\sigma$ ratio in terms of $m$, $n$, $k_m$, and $k_n$. The first threshold, $\sf SNR_c$, corresponds to the computational boundary. Below this threshold, it is shown that no polynomial time algorithm can succeed in identifying the submatrix, under the \textit{hidden clique hypothesis}. We introduce adaptive linear time spectral algorithms that identify the submatrix with high probability when the signal strength is above the threshold $\sf SNR_c$. The second threshold, $\sf SNR_s$, captures the statistical boundary, below which no method can succeed with probability going to one in the minimax sense. The exhaustive search method successfully finds the submatrix above this threshold. The results show an interesting phenomenon that $\sf SNR_c$ is always significantly larger than $\sf SNR_s$, which implies an essential gap between statistical optimality and computational efficiency for submatrix localization.