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13 - Statistical Problems with Planted Structures: Information-Theoretical and Computational Limits

Published online by Cambridge University Press:  22 March 2021

By
Yihong Wu  and
Jiaming Xu
Edited by
Miguel R. D. Rodrigues  and
Yonina C. Eldar
Miguel R. D. Rodrigues
Affiliation:
University College London
Yonina C. Eldar
Affiliation:
Weizmann Institute of Science, Israel
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Summary

This chapter provides a survey of the common techniques for determining the sharp statistical and computational limits in high-dimensional statistical problems with planted structures, using community detection and submatrix detection problems as illustrative examples. We discuss tools including the first- and second-moment methods for analyzing the maximum-likelihood estimator, information-theoretic methods for proving impossibility results using mutual information and rate-distortion theory, and methods originating from statistical physics such as the interpolation method. To investigate computational limits, we describe a common recipe to construct a randomized polynomial-time reduction scheme that approximately maps instances of the planted clique problem to the problem of interest in total variation distance.

Keywords

statistical limitscomputational limitsstatistical computational trade-offsphase transitioncommunity detectioncommunity recoverysubmatrix detectionsubmatrix recovery
Type
Chapter
Information
Information-Theoretic Methods in Data Science , pp. 383 - 424
Publisher: Cambridge University Press
Print publication year: 2021

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