2013/01/20 by Cristina Butucea, Butucea, Cristina, Ghislaine Gayraud +1
Mathematics · #60C20 #60G15 #62G10 #62G20 #62H15 #FOS: Mathematics #Mathematical Approximation and Integration #Random Matrices and Applications #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #msc:60C20 #msc:60G15 #msc:62G10 #msc:62G20 #msc:62H15 #stat.TH
paper · pdf · doi:10.48550/arxiv.1301.4660
arxiv created 2013/01/20 · openalex publication_date 2013/01/20 · arxiv updated 2013/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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,k for i ∈\1,...,M\, j ∈ \1,...,N\ and k ∈ ℤ. The standard deviation of our observations is \ep ks for some \ep >0 and s ≥ 0. We give sharp rates for the detection of a sparse submatrix of size m × n with active components. A component (i,j) is said active if the sequence \xij,k\k have mean \θij,k\k within a Sobolev ellipsoid of smoothness τ>0 and total energy ∑k θ2ij,k larger than some r2_\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.