The paper studies the signal detection problem in sparse additive models, where the goal is to determine the smallest rate ε as a function of the sparsity level s, the dimension p, the sample size n, and the function space H such that consistent detection of the alternative against the null is possible.
The key findings are:
The nonasymptotic minimax separation rate exhibits a nontrivial interaction between the sparsity level s and the choice of function space H. This is in contrast to the estimation theory, where the minimax rate is separable into a sparsity term and a function space term.
An adaptive testing rate is established for a generic function space H. The adaptation cost depends delicately on the properties of H, extending previous results that focused on Sobolev spaces.
In the Sobolev space setting, the paper corrects some existing claims in the literature regarding adaptation to both sparsity and smoothness.
The paper develops a nonasymptotic minimax testing framework and employs thresholded χ^2 statistics as the testing procedure. The analysis reveals a phase transition in the separation rate depending on the interaction between the sparsity level and the function space.
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by Subhodh Kote... at arxiv.org 10-03-2024
https://arxiv.org/pdf/2304.09398.pdfDeeper Inquiries