Iterative reconstruction of rank-one matrices in noise

Alyson K. Fletcher, Sundeep Rangan

Research output: Contribution to journalArticle

Abstract

We consider the problem of estimating a rank-one matrix in Gaussian noise under a probabilistic model for the left and right factors of the matrix. The probabilistic model can impose constraints on the factors including sparsity and positivity that arise commonly in learning problems. We propose a family of algorithms that reduce the problem to a sequence of scalar estimation computations. These algorithms are similar to approximate message-passing techniques based on Gaussian approximations of loopy belief propagation that have been used recently in compressed sensing. Leveraging analysis methods by Bayati and Montanari, we show that the asymptotic behavior of the algorithm is described by a simple scalar equivalent model, where the distribution of the estimates at each iteration is identical to certain scalar estimates of the variables in Gaussian noise. Moreover, the effective Gaussian noise level is described by a set of state evolution equations. The proposed approach to deriving algorithms thus provides a computationally simple and general method for rank-one estimation problems with a precise analysis in certain high-dimensional settings.

Original languageEnglish (US)
Pages (from-to)531-562
Number of pages32
JournalInformation and Inference
Volume7
Issue number3
DOIs
StatePublished - 2018

Keywords

  • Approximate message passing. 2000 Math Subject Classification: 34K30, 35K57, 35Q80, 92D25
  • Bayesian estimation
  • Matrix factorization

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Analysis
  • Applied Mathematics
  • Statistics and Probability
  • Numerical Analysis

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