2015/02/02 by Olivier Collier, Collier, Olivier, Laëtitia Comminges +3 · 3 citations
Engineering · Mathematics · #FOS: Mathematics #Mathematical Approximation and Integration #Sparse and Compressive Sensing Techniques #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1502.00665
openalex publication_date 2015/02/02 · openalex created_date 2022/11/26 · openalex updated_date 2026/07/28
For the Gaussian sequence model, we obtain non-asymptotic minimax rates of\nestimation of the linear, quadratic and the L2-norm functionals on classes of\nsparse vectors and construct optimal estimators that attain these rates. The\nmain object of interest is the class s-sparse vectors for which we also provide\ncompletely adaptive estimators (independent of s and of the noise variance)\nhaving only logarithmically slower rates than the minimax ones. Furthermore, we\nobtain the minimax rates on the Lq-balls where 0 < q < 2. This analysis shows\nthat there are, in general, three zones in the rates of convergence that we\ncall the sparse zone, the dense zone and the degenerate zone, while a fourth\nzone appears for estimation of the quadratic functional. We show that, as\nopposed to estimation of the vector, the correct logarithmic terms in the\noptimal rates for the sparse zone scale as log(d/s2) and not as log(d/s). For\nthe sparse class, the rates of estimation of the linear functional and of the\nL2-norm have a simple elbow at s = sqrt(d) (boundary between the sparse and the\ndense zones) and exhibit similar performances, whereas the estimation of the\nquadratic functional reveals more complex effects and is not possible only on\nthe basis of sparsity described by the sparsity condition on the vector.\nFinally, we apply our results on estimation of the L2-norm to the problem of\ntesting against sparse alternatives. In particular, we obtain a non-asymptotic\nanalog of the Ingster-Donoho-Jin theory revealing some effects that were not\ncaptured by the previous asymptotic analysis.\n