Minimax estimation of linear and quadratic functionals on sparsity classes

Abstract

For the Gaussian sequence model, we obtain non-asymp-totic minimax rates of estimation of the linear, quadratic and the ℓ 2 -norm functionals on classes of sparse vectors and construct optimal estimators that attain these rates. The main object of interest is the class B 0 (s) of s-sparse vectors θ = (θ 1 , . . . , θ d ), for which we also provide completely adaptive estimators (independent of s and of the noise variance σ) having only logarithmically slower rates than the minimax ones. Furthermore, we obtain the minimax rates on. This analysis shows that there are, in general, three zones in the rates of convergence that we call the sparse zone, the dense zone and the degenerate zone, while a fourth zone appears for estimation of the quadratic functional. We show that, as opposed to estimation of θ, the correct logarithmic terms in the optimal rates for the sparse zone scale as log(d/s 2 ) and not as log(d/s). For the class B 0 (s), the rates of estimation of the linear functional and of the ℓ 2 -norm have a simple elbow at s = √ d (boundary between the sparse and the dense zones) and exhibit similar performances, whereas the estimation of the quadratic functional Q(θ) reveals more complex effects and is not possible only on the basis of sparsity described by the condition θ ∈ B 0 (s). Finally, we apply our results on estimation of the ℓ 2 -norm to the problem of testing against sparse alternatives. In particular, we obtain a non-asymptotic analog of Ingster-Donoho-Jin theory revealing some effects that were not captured by the previous asymptotic analysis.

Minimax Rate of Testing in Sparse Linear Regression

“…Note that the rate λ defined in this way is a non-asymptotic minimax rate of testing as opposed to the classical asymptotic definition that can be found, for example, in Ingster and Suslina (2003). It is shown in Collier, Comminges and Tsybakov (2017) that when X is the identity matrix and p = n (which corresponds to the Gaussian sequence model), the non-asymptotic minimax rate of testing on the class B 0 (s) with respect to the ℓ 2 -distance has the following form:…”

Adaptive estimation of the sparsity in the Gaussian vector model

“…The signal detection problem which amounts to testing whether θ = 0 is a special instance of the sparsity testing problem (corresponding to k 0 = 0). Signal detection in the Gaussian vector model has been extensively studied [3,14,17,25] in the last fifteen years and is now well understood. For instance, it has been established in [14] that the minimax separation distance ρ * γ [0, ∆] satisfies…”

“…This entails that it is possible to detect sparse vectors θ whose non-zero values are of order log(n/∆ 2 ). Known optimal tests such as the higher criticism test [17] or the one proposed in [14] amount to counting the number of values |Y i | that are larger than t and to compare this number to what is expected under the null hypothesis. Doing this simultaneously for a wide range of t leads to near-optimal performances simultaneously for all ∆ ∈ [1, √ n].…”

Rate optimal estimation of quadratic functionals in inverse problems with partially unknown operator and application to testing problems

“…-Finally, considering inverse problems with sparsity constraints as in [CCT17] might be of interest.…”