## EM-BG-GAMP: An Algorithm for Sparse Reconstruction

### Overview

The Expectation-Maximization Bernoulli-Gaussian Approximate Message Passing (EM-BG-GAMP) is an algorithm designed to recover a signal $$\boldsymbol{x} \in \mathbb{R}^N$$ from (possibly) noisy measurements $$\boldsymbol{y} = \boldsymbol{Ax} + \boldsymbol{w} \in \mathbb{R}^M$$ . One particularly interesting case is when the measurements are undersampled (i.e. $$M \lt N$$). With sufficient signal sparsity and a well-conditioned mixing matrix, signal recovery is possible.

EM-BG-GAMP assumes a sparsity promoting i.i.d. Bernoulli-Gaussian prior: $$p(x_n) = (1-\lambda)\delta(x_n) + \lambda \mathcal{N}(x_n; \theta,\phi)$$ and i.i.d. additive Gaussian noise: $$p(w_m) = \mathcal{N}(w_m;0,\psi)$$ . It then uses the Generalized Approximate Message Passing (GAMP) algorithm to evaluate the means (MMSE estimates) and variances of the posterior $$p(\boldsymbol{x}|\boldsymbol{y})$$. After running BG-GAMP, we then find the Expectation-Maximization (EM) updates of the parameters $$\boldsymbol{q} \triangleq [\lambda, \theta, \phi, \psi]$$ and repeat BG-GAMP until convergence. (See publications below for details.)

The EM-BG-GAMP MATLAB function attempts to recover a signal through measurements observed from the above linear model. Unlike some other compressive sensing algorithms (i.e. LASSO), EM-BG-GAMP does not require any tuning parameters. EM-BG-GAMP does, however, have some optional inputs to improve computation time/accuracy. Currently, EM-BG-GAMP supports both real and complex signals. Examples of the function's usage can be seen in the Examples section.

There are plenty of compressive-sensing algorithms available. Why use EM-BG-GAMP? Here are a few advantages of our approach.

1. Excellent recovery performance over a wide range of signal/matrix types.
2. Very good complexity scaling with problem dimensions
3. No required tuning parameters.

### Publications

The details are in:

#### Journal

1. J. P. Vila and P. Schniter, Expectation-Maximization Gaussian-Mixture Approximate Message Passing [pdf][arxiv],'' IEEE Transactions on Signal Processing, vol. 61, no. 19, pp. 4658-4672, Oct. 2013.

#### Conference

1. J. P. Vila and P. Schniter, Expectation-Maximization Bernoulli-Gaussian Approximate Message Passing [pdf],'' Proc. Asilomar Conf. on Signals, Systems, and Computers (Pacific Grove, CA), Nov. 2011. [slides]

#### Poster

1. J. P. Vila and P. Schniter, An Empirical-Bayes Approach to Compressive Sensing via Approximate Message Passing [pdf],'' presented at the Duke Workshop on Sensing and Analysis of High-Dimensional Data (SAHD) (Durham, North Carolina), July 2011. [poster]

### Support

This work has been supported in part by NSF-I/UCRC grant IIP-0968910, by NSF grant CCF-1018368, and by DARPA/ONR grant N66001-10-1-4090.

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