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Activity Number: 345 - High-Dimensional Statistics
Type: Contributed
Date/Time: Tuesday, July 30, 2019 : 10:30 AM to 12:20 PM
Sponsor: IMS
Abstract #306759
Title: Fundamental Limits of Exact Support Recovery in High Dimensions
Author(s): Zheng Gao* and Stilian Stoev
Companies: University of Michigan and University of Michigan
Keywords: support recovery; high–dimensional inference; relative stability; rapid variation; concentration of maxima

We study the support recovery problem for a high-dimensional signal observed with additive noise. With suitable parametrization of the signal sparsity and magnitude of its non-zero components, we characterize a phase-transition phenomenon akin to the signal detection problem studied by Ingster in 1998. Specifically, if the signal is above the so-called strong classification boundary, we show that several classes of well-known procedures achieve asymptotically perfect support recovery as the dimension goes to infinity. This is so, for a very broad class of error distributions with light, rapidly varying tails which may have arbitrary dependence. Conversely, if the signal is below the boundary, then for a very broad class of error dependence structures, no thresholding estimators (including data-dependent procedures) can achieve perfect support recovery.

The boundary is established for dependent errors in a fine, point-wise, rather than minimax, sense. Its proof exploits a certain concentration of maxima phenomenon known as relative stability. We provide a complete characterization of the relative stability phenomenon for Gaussian arrays in terms their correlation structures.

Authors who are presenting talks have a * after their name.

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