Orthogonal Matching Pursuit: A Brownian Motion Analysis

A well-known analysis of Tropp and Gilbert shows that orthogonal matching pursuit (OMP) can recover a k-sparse n-dimensional real vector from 4 k log(n) noise-free linear measurements obtained through a random Gaussian measurement matrix with a probability that approaches one as n approaches infinity. This work strengthens this result by showing that a lower number of measurements, 2 k log(n - k), is in fact sufficient for asymptotic recovery. More generally, when the sparsity level satisfies kmin <= k <= kmax but is unknown, 2 kmax log(n - kmin) measurements is sufficient. Furthermore, this number of measurements is also sufficient for detection of the sparsity pattern (support) of the vector with measurement errors provided the signal-to-noise ratio (SNR) scales to infinity. The scaling 2 k log(n - k) exactly matches the number of measurements required by the more complex lasso method for signal recovery with a similar SNR scaling.Comment: 11 pages, 2 figure

Vector Approximate Message Passing for the Generalized Linear Model

The generalized linear model (GLM), where a random vector $\boldsymbol{x}$ is observed through a noisy, possibly nonlinear, function of a linear transform output $\boldsymbol{z}=\boldsymbol{Ax}$, arises in a range of applications such as robust regression, binary classification, quantized compressed sensing, phase retrieval, photon-limited imaging, and inference from neural spike trains. When $\boldsymbol{A}$ is large and i.i.d. Gaussian, the generalized approximate message passing (GAMP) algorithm is an efficient means of MAP or marginal inference, and its performance can be rigorously characterized by a scalar state evolution. For general $\boldsymbol{A}$, though, GAMP can misbehave. Damping and sequential-updating help to robustify GAMP, but their effects are limited. Recently, a "vector AMP" (VAMP) algorithm was proposed for additive white Gaussian noise channels. VAMP extends AMP's guarantees from i.i.d. Gaussian $\boldsymbol{A}$ to the larger class of rotationally invariant $\boldsymbol{A}$. In this paper, we show how VAMP can be extended to the GLM. Numerical experiments show that the proposed GLM-VAMP is much more robust to ill-conditioning in $\boldsymbol{A}$ than damped GAMP

Asymptotic Analysis of MAP Estimation via the Replica Method and Applications to Compressed Sensing

The replica method is a non-rigorous but well-known technique from statistical physics used in the asymptotic analysis of large, random, nonlinear problems. This paper applies the replica method, under the assumption of replica symmetry, to study estimators that are maximum a posteriori (MAP) under a postulated prior distribution. It is shown that with random linear measurements and Gaussian noise, the replica-symmetric prediction of the asymptotic behavior of the postulated MAP estimate of an n-dimensional vector "decouples" as n scalar postulated MAP estimators. The result is based on applying a hardening argument to the replica analysis of postulated posterior mean estimators of Tanaka and of Guo and Verdu. The replica-symmetric postulated MAP analysis can be readily applied to many estimators used in compressed sensing, including basis pursuit, lasso, linear estimation with thresholding, and zero norm-regularized estimation. In the case of lasso estimation the scalar estimator reduces to a soft-thresholding operator, and for zero norm-regularized estimation it reduces to a hard-threshold. Among other benefits, the replica method provides a computationally-tractable method for precisely predicting various performance metrics including mean-squared error and sparsity pattern recovery probability.Comment: 22 pages; added details on the replica symmetry assumptio

Inference for Generalized Linear Models via Alternating Directions and Bethe Free Energy Minimization

Generalized Linear Models (GLMs), where a random vector $\mathbf{x}$ is observed through a noisy, possibly nonlinear, function of a linear transform $\mathbf{z}=\mathbf{Ax}$ arise in a range of applications in nonlinear filtering and regression. Approximate Message Passing (AMP) methods, based on loopy belief propagation, are a promising class of approaches for approximate inference in these models. AMP methods are computationally simple, general, and admit precise analyses with testable conditions for optimality for large i.i.d. transforms $\mathbf{A}$. However, the algorithms can easily diverge for general $\mathbf{A}$. This paper presents a convergent approach to the generalized AMP (GAMP) algorithm based on direct minimization of a large-system limit approximation of the Bethe Free Energy (LSL-BFE). The proposed method uses a double-loop procedure, where the outer loop successively linearizes the LSL-BFE and the inner loop minimizes the linearized LSL-BFE using the Alternating Direction Method of Multipliers (ADMM). The proposed method, called ADMM-GAMP, is similar in structure to the original GAMP method, but with an additional least-squares minimization. It is shown that for strictly convex, smooth penalties, ADMM-GAMP is guaranteed to converge to a local minima of the LSL-BFE, thus providing a convergent alternative to GAMP that is stable under arbitrary transforms. Simulations are also presented that demonstrate the robustness of the method for non-convex penalties as well