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

We consider a matrix-valued Gaussian sequence model, that is, we observe a sequence of high-dimensional M ×N matrices of heterogeneous Gaussian random variables xij,κ for i ∈ {1,. ., M}, j ∈ {1,. ., N}, M and N tend to infinity and κ ∈ Z. For large |κ|, the standard deviation of our observations is ∈|κ|s for some ∈ > 0, ∈ →0 and a given s ≥ 0, case that encompasses mildly ill-posed inverse problems. We give separation rates for the detection of a sparse submatrix of size m × n (m and n tend to infinity, m/M and n/N tend 0) with active components. A component (i, j ) is said active if the sequence {x_ij,κ } κ has mean {θ _ij,κ } κ within a Sobolev ellipsoid of smoothness τ >0 and total energy ∑ κ θ _ij,κ ² larger than some r² ∈. We construct a test procedure and compute rates that involve relationships between m,n,M, N and ∈, such that the asymptotic total error probability tends to 0. We also show how these rates can be made adaptive to the size (m, n) of the submatrix under some constraints. We prove corresponding lower bounds under additional assumptions on the relative size of the submatrix in the large matrix of observations. Our separation rates are sharp under further assumptions. 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 total error probability for testing tends to 1.