Guaranteed sparse recovery under linear transformation

Ji Liu, Lei Yuan, Jieping Ye

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

3 Scopus citations

Abstract

We consider the following signal recovery problem: given a measurement matrix Φ ∈ ℝnxp and a noisy observation vector c ∈ ℝnconstructed from c = Φθ*+ ε where ε ∈ ℝn is the noise vector whose entries follow i.i.d. centered sub-Gaussian distribution, how to recover the signal θ&z,ast; if Dθ*is sparse under a linear transformation D ∈ ℝmxp? One natural method using convex optimization is to solve the following problem:(Equation Presented) This paper provides an upper bound of the estimate error and shows the consistency property of this method by assuming that the design matrix $ is a Gaussian random matrix. Specifically, we show 1) in the noiseless case, if the condition number of D is bounded and the measurement number n ≥ Ω(slog(p)) where s is the sparsity number, then the true solution can be recovered with high probability; and 2) in the noisy case, if the condition number of D is bounded and the measurement increases faster than slog(p), that is, slog(p) = o(n), the estimate error converges to zero with probability 1 when p and s go to infinity. Our resuits are consistent with those for the special case D = Ipxp (equivalently LASSO) and improve the existing analysis. The condition number of D plays a critical role in our analysis. We consider the condition numbers in two cases including the fused LASSO and the random graph: the condition number in the fused LASSO case is bounded by a constant, while the condition number in the random graph case is bounded with high probability if m/p (i.e., #edge/#vertex) is larger than a certain constant. Numerical simulations are consistent with our theoretical results.

Original languageEnglish (US)
Title of host publication30th International Conference on Machine Learning, ICML 2013
PublisherInternational Machine Learning Society (IMLS)
Pages1128-1136
Number of pages9
EditionPART 2
StatePublished - 2013
Event30th International Conference on Machine Learning, ICML 2013 - Atlanta, GA, United States
Duration: Jun 16 2013Jun 21 2013

Other

Other30th International Conference on Machine Learning, ICML 2013
CountryUnited States
CityAtlanta, GA
Period6/16/136/21/13

ASJC Scopus subject areas

  • Human-Computer Interaction
  • Sociology and Political Science

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  • Cite this

    Liu, J., Yuan, L., & Ye, J. (2013). Guaranteed sparse recovery under linear transformation. In 30th International Conference on Machine Learning, ICML 2013 (PART 2 ed., pp. 1128-1136). International Machine Learning Society (IMLS).