Sharp detection of smooth signals in a high-dimensional sparse matrix with indirect observations

TL;DR

This paper establishes sharp detection rates for sparse, smooth signals in high-dimensional Gaussian matrices with indirect, heteroscedastic observations, advancing understanding of signal detectability in complex data models.

Contribution

It provides the first sharp asymptotic detection rates for sparse, smooth signals in high-dimensional matrices with indirect Gaussian observations, including minimax lower bounds.

Findings

Derived sharp detection rates for sparse submatrix signals.

Established asymptotic minimax risk tending to zero under certain conditions.

Proved lower bounds showing no test can outperform the established rates.

Abstract

We consider a matrix-valued Gaussian sequence model, that is, we observe a sequence of high-dimensional $M \times N$ matrices of heterogeneous Gaussian random variables $x_{ij,k}$ for $i \in\{1,...,M\}$, $j \in \{1,...,N\}$ and $k \in \mathbb{Z}$. The standard deviation of our observations is $\ep k^s$ for some $\ep >0$ and $s \geq 0$. We give sharp rates for the detection of a sparse submatrix of size $m \times n$ with active components. A component $(i,j)$ is said active if the sequence $\{x_{ij,k}\}_k$ have mean $\{\theta_{ij,k}\}_k$ within a Sobolev ellipsoid of smoothness $\tau >0$ and total energy $\sum_k \theta^2_{ij,k} $ larger than some $r^2_\ep$. Our rates involve relationships between $m,\, n, \, M$ and $N$ tending to infinity such that $m/M$, $n/N$ and $\ep$ tend to 0, such that a test procedure that we construct has asymptotic minimax risk tending to 0. We prove corresponding lower bounds under additional assumptions on the relative size of the submatrix in the large matrix of observations. Except for these additional conditions our rates are asymptotically sharp. Lower bounds for hypothesis testing problems mean that no test procedure can distinguish between the null hypothesis (no signal) and the alternative, i.e. the minimax risk for testing tends to 1.