Resolution limits of sparse coding in high dimensions

Alyson K. Fletcher, Sundeep Rangan, Vivek K. Goyal

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

This paper addresses the problem of sparsity pattern detection for unknown κ-sparse n-dimensional signals observed through m noisy, random linear measurements. Sparsity pattern recovery arises in a number of settings including statistical model selection, pattern detection, and image acquisition. The main results in this paper are necessary and sufficient conditions for asymptotically-reliable sparsity pattern recovery in terms of the dimensions m, n and k as well as the signal-tonoise ratio (SNR) and the minimum-to-average ratio (MAR) of the nonzero entries of the signal. We show that m > 2κ log(n - κ)/(SNR ?MAR) is necessary for any algorithm to succeed, regardless of complexity; this matches a previous sufficient condition for maximum likelihood estimation within a constant factor under certain scalings of κ, SNR and MAR with n. We also show a sufficient condition for a computationally-trivial thresholding algorithm that is larger than the previous expression by only a factor of 4(1+SNR) and larger than the requirement for lasso by only a factor of 4/MAR. This provides insight on the precise value and limitations of convex programming-based algorithms.

Original languageEnglish (US)
Title of host publicationAdvances in Neural Information Processing Systems 21 - Proceedings of the 2008 Conference
Pages449-456
Number of pages8
StatePublished - 2009
Event22nd Annual Conference on Neural Information Processing Systems, NIPS 2008 - Vancouver, BC, Canada
Duration: Dec 8 2008Dec 11 2008

Publication series

NameAdvances in Neural Information Processing Systems 21 - Proceedings of the 2008 Conference

Other

Other22nd Annual Conference on Neural Information Processing Systems, NIPS 2008
CountryCanada
CityVancouver, BC
Period12/8/0812/11/08

ASJC Scopus subject areas

  • Information Systems

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  • Cite this

    Fletcher, A. K., Rangan, S., & Goyal, V. K. (2009). Resolution limits of sparse coding in high dimensions. In Advances in Neural Information Processing Systems 21 - Proceedings of the 2008 Conference (pp. 449-456). (Advances in Neural Information Processing Systems 21 - Proceedings of the 2008 Conference).