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

Understand

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 $r^2_\ep$.