# Message passing algorithms

I see these popping up in graphical model inference, in time-series, and in variational approximation and compressed sensing. What are they?

The grandparent idea seems to be “Belief propagation”, a.k.a. “sum-product message-passing”, credited to (Pearl, 1982) for DAGs and then generalised to MRFs, PGMs, factor graphs etc. Although I gather from passing reference that many popoular algorithms also happen to be message-passing-type ones.

Now there are many flavours, such as “Gaussian” and “approximate”. Apparently this definition subsumes such diverse models as the Viterbi and Baum-Welch algorithms, among others. See WaJo08 for an overview.

Anyway, what the hell are these things?

The recipe to make a message-passing algorithm has four steps:

1. Pick an approximating family for q to be chosen from. For example, the set of fully-factorized distributions, the set of Gaussians, the set of k-component mixtures, etc.
2. Pick a divergence measure to minimize. For example, mean-field methods minimize the Kullback-Leibler divergence $KL(q \| p)$, expectation propagation minimizes $KL(p \| q)$, and power EP minimizes α-divergence, $D\alpha(p \| q)$.
3. Construct an optimization algorithm for the chosen divergence measure and approximating family. Usually this is a fixed-point iteration obtained by setting the gradients to zero.
4. Distribute the optimization across the network, by dividing the network p into factors, and minimizing local divergence at each factor.

## Grandiose Claims

• GAMP

Generalized Approximate Message Passing (GAMP) is an approximate, but computationally efficient method for estimation problems with linear mixing. In the linear mixing problem an unknown vector, $\mathbf{x}$, with independent components, is first passed through linear transform $\mathbf{z}=\mathbf{A}\mathbf{x}$ and then observed through a general probabilistic, componentwise measurement channel to yield a measurement vector $\mathbf{y}$. The problem is to estimate $\mathbf{x}$ and $\mathbf{z}$ from $\mathbf{y}$ and $\mathbf{A}$. This problem arises in a range of applications including compressed sensing.

Optimal solutions to linear mixing estimation problems are, in general, computationally intractable as the complexity of most brute force algorithms grows exponentially in the dimension of the vector $\mathbf{x}$. GAMP approximately performs the estimation through a Gaussian approximation of loopy belief propagation that reduces the vector-valued estimation problem to a sequence of scalar estimation problems on the components of the vectors $\mathbf{x}$ and $\mathbf{z}$. The GAMP methodology may also have applications to problems with structured sparsity and low-rank matrix factorization

BaMo11
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DoMM09a
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DoMM09b
Donoho, D. L., Maleki, A., & Montanari, A. (2009b) Message passing algorithms for compressed sensing: II analysis and validation. In 2010 IEEE Information Theory Workshop (ITW) (pp. 1–5). DOI.
DoMM10
Donoho, D. L., Maleki, A., & Montanari, A. (2010) Message passing algorithms for compressed sensing: I motivation and construction. In 2010 IEEE Information Theory Workshop (ITW) (pp. 1–5). DOI.
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