Sparse Signal Detection in Heteroscedastic Gaussian Sequence Models: Sharp Minimax Rates
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Given a heterogeneous Gaussian sequence model with unknown mean θ∈ℝ^d and known covariance matrix Σ = diag(σ_1^2,…, σ_d^2), we study the signal detection problem against sparse alternatives, for known sparsity s. Namely, we characterize how large ϵ^*>0 should be, in order to distinguish with high probability the null hypothesis θ=0 from the alternative composed of s-sparse vectors in ℝ^d, separated from 0 in L^t norm (t ≥ 1) by at least ϵ^*. We find minimax upper and lower bounds over the minimax separation radius ϵ^* and prove that they are always matching. We also derive the corresponding minimax tests achieving these bounds. Our results reveal new phase transitions regarding the behavior of ϵ^* with respect to the level of sparsity, to the L^t metric, and to the heteroscedasticity profile of Σ. In the case of the Euclidean (i.e. L^2) separation, we bridge the remaining gaps in the literature.
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