Approximate message passing with consistent parameter estimation and applications to sparse learning

Ulugbek S. Kamilov, Sundeep Rangan, Alyson K. Fletcher, Michael Unser

Research output: Contribution to journalArticlepeer-review


We consider the estimation of an independent and identically distributed (i.i.d.) (possibly non-Gaussian) vector {\bf x}\in{\BBR}{n} from measurements {\bf y}\in{\BBR}{m} obtained by a general cascade model consisting of a known linear transform followed by a probabilistic componentwise (possibly nonlinear) measurement channel. A novel method, called adaptive generalized approximate message passing (adaptive GAMP) is presented. It enables the joint learning of the statistics of the prior and measurement channel along with estimation of the unknown vector {\bf x}. We prove that, for large i.i.d. Gaussian transform matrices, the asymptotic componentwise behavior of the adaptive GAMP is predicted by a simple set of scalar state evolution equations. In addition, we show that the adaptive GAMP yields asymptotically consistent parameter estimates, when a certain maximum-likelihood estimation can be performed in each step. This implies that the algorithm achieves a reconstruction quality equivalent to the oracle algorithm that knows the correct parameter values. Remarkably, this result applies to essentially arbitrary parametrizations of the unknown distributions, including nonlinear and non-Gaussian ones. The adaptive GAMP methodology thus provides a systematic, general and computationally efficient method applicable to a large range of linear-nonlinear models with provable guarantees.

Original languageEnglish (US)
Article number6775335
Pages (from-to)2969-2985
Number of pages17
JournalIEEE Transactions on Information Theory
Issue number5
StatePublished - May 2014


  • Approximate message passing
  • belief propagation
  • compressive sensing
  • parameter estimation

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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