[Paper Review] Approximate Message Passing with Consistent Parameter Estimation and Applications to Sparse Learning

Ulugbek S. Kamilov, Sundeep Rangan+2 more|arXiv (Cornell University)|Jul 17, 2012

Blind Source Separation Techniques33 references14 citations

TL;DR

This paper introduces adaptive generalized approximate message passing (Adaptive GAMP), a computationally efficient algorithm that jointly estimates sparse signals and learns unknown parameters in linear-nonlinear models. It proves asymptotic consistency of parameter estimates via state evolution, ensuring performance equivalent to oracle methods even for non-Gaussian, nonlinear models.

ABSTRACT

We consider the estimation of an i.i.d. (possibly non-Gaussian) vector $\xbf \in \R^n$ from measurements $\ybf \in \R^m$ 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), that enables joint learning of the statistics of the prior and measurement channel along with estimation of the unknown vector $\xbf$ is presented. The proposed algorithm is a generalization of a recently-developed EM-GAMP that uses expectation-maximization (EM) iterations where the posteriors in the E-steps are computed via approximate message passing. The methodology can be applied to a large class of learning problems including the learning of sparse priors in compressed sensing or identification of linear-nonlinear cascade models in dynamical systems and neural spiking processes. We prove that for large i.i.d. Gaussian transform matrices the asymptotic componentwise behavior of the adaptive GAMP algorithm is predicted by a simple set of scalar state evolution equations. In addition, we show that when a certain maximum-likelihood estimation can be performed in each step, the adaptive GAMP method can yield asymptotically consistent parameter estimates, which 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 ones that are nonlinear and non-Gaussian. The adaptive GAMP methodology thus provides a systematic, general and computationally efficient method applicable to a large range of complex linear-nonlinear models with provable guarantees.

Motivation & Objective

• Address the challenge of joint estimation of sparse signals and unknown parameters in linear-nonlinear models where prior and noise distributions are not fully known.
• Develop a general, computationally efficient method applicable to diverse problems such as compressed sensing and neural spike response modeling.
• Provide theoretical guarantees for parameter estimation consistency under large i.i.d. Gaussian transforms.
• Unify and generalize existing EM-GAMP approaches by embedding maximum-likelihood parameter adaptation into the message-passing framework.
• Establish a rigorous asymptotic analysis using scalar state evolution equations to predict algorithm behavior.

Proposed method

• Propose Adaptive GAMP, a generalization of GAMP that integrates expectation-maximization (EM) with approximate message passing for joint parameter and signal estimation.
• Use iterative adaptation functions $ H_x^t $ and $ H_z^t $ to estimate unknown parameters $ heta_x $ and $ heta_z $ of the prior and measurement channel using maximum-likelihood principles.
• Derive scalar state evolution (SE) equations to describe the asymptotic componentwise behavior of the algorithm under large i.i.d. Gaussian transform matrices.
• Ensure consistency of parameter estimates by proving that the adaptation functions converge to true parameters under mild regularity conditions on the likelihood functions.
• Apply pseudo-Lipschitz continuity and compactness assumptions to establish convergence of parameter estimates to their true values in the large-system limit.
• Leverage the structure of the state evolution to show that the algorithm achieves performance equivalent to an oracle that knows the true parameters.

Experimental results

Research questions

• RQ1 Can a message-passing algorithm jointly estimate sparse signals and learn unknown parameters in linear-nonlinear models with provable consistency?
• RQ2 Does the proposed Adaptive GAMP method achieve asymptotic performance equivalent to an oracle algorithm with full knowledge of the true parameters?
• RQ3 Can the state evolution framework be extended to analyze adaptive parameter estimation in AMP-type algorithms?
• RQ4 Under what conditions does the adaptive parameter estimation in GAMP converge to the true parameter values?
• RQ5 How does the algorithm perform for non-Gaussian and nonlinear measurement models?

Key findings

• Adaptive GAMP achieves asymptotically consistent parameter estimates for both prior and measurement channel parameters under mild regularity conditions.
• The asymptotic componentwise behavior of Adaptive GAMP is exactly characterized by a set of scalar state evolution equations, extending classical AMP analysis to adaptive settings.
• When the adaptation functions perform maximum-likelihood estimation at each iteration, the algorithm achieves performance equivalent to an oracle that knows the true parameters.
• The consistency result holds for arbitrary parametrizations of the distributions, including nonlinear and non-Gaussian models.
• The convergence of parameter estimates to their true values is proven via continuity and uniqueness of the maximum-likelihood solution in the large-system limit.
• The method generalizes existing EM-GAMP approaches and provides a unified framework for learning in sparse linear models with provable guarantees.