This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and concepts–in particular making ties to topics such as wavelets and multigrid methods. A third is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for self-similar and 1/f processes. We also illustrate how these methods have been used in practice. We discuss the construction of MR models on trees and show how questions that arise in this context make contact with wavelets, state space modeling of time series, system and parameter identification, and hidden

We present a tree-based reparameterization framework that provides a new conceptual view of a large class of algorithms for computing approximate marginals in graphs with cycles. This class includes the belief propagation or sum-product algorithm [39, 36], as well as a rich set of variations and extensions of belief propagation. Algorithms in this class can be formulated as a sequence of reparameterization updates, each of which entails re-factorizing a portion of the distribution corresponding to an acyclic subgraph (i.e., a tree). The ultimate goal is to obtain an alternative but equivalent factorization using functions that represent (exact or approximate) marginal distributions on cliques of the graph. Our framework highlights an important property of BP and the entire class of reparameterization algorithms: the distribution on the full graph is not changed. The perspective of tree-based updates gives rise to a simple and intuitive characterization of the fixed points in terms of tree consistency. We develop interpretations of these results in terms of information geometry. The invariance of the distribution, in conjunction with the fixed point characterization, enables us to derive an exact relation between the exact marginals on an arbitrary graph with cycles, and the approximations provided by belief propagation, and more broadly, any algorithm that minimizes the Bethe free energy. We also develop bounds on this approximation error, which illuminate the conditions that govern their accuracy. Finally, we show how the reparameterization perspective extends naturally to more structured approximations (e.g., Kikuchi and variants [52, 37]) that operate over higher order cliques.

We present a new framework based on walks in a graph for analysis and inference in Gaussian graphical models. The key idea is to decompose the correlation between each pair of variables as a sum over all walks between those variables in the graph. The weight of each walk is given by a product of edge-wise partial correlation coefficients. This representation holds for a large class of Gaussian graphical models which we call walk-summable. We give a precise characterization of this class of models, and relate it to other classes including diagonally dominant, attractive, nonfrustrated, and pairwise-normalizable. We provide a walk-sum interpretation of Gaussian belief propagation in trees and of the approximate method of loopy belief propagation in graphs with cycles. The walk-sum perspective leads to a better understanding of Gaussian belief propagation and to stronger results for its convergence in loopy graphs.

Graphical models are well suited to capture the complex and non-Gaussian statistical dependencies that arise in many real-world signals. A fundamental problem common to any signal processing application of a graphical model is that of computing approximate marginal probabilities over subsets of nodes. This paper proposes a novel method, applicable to discrete-valued Markov random fields (MRFs) on arbitrary graphs, for approximately solving this marginalization problem. The foundation of our method is a reformulation of the marginalization problem as the solution of a low-dimensional convex optimization problem over the marginal polytope. Exactly solving this problem for general graphs is intractable; for binary Markov random fields, we describe how to relax it by using a Gaussian bound on the discrete entropy and a semidefinite outer bound on the marginal polytope. This combination leads to a log-determinant maximization problem that can be solved efficiently by interior point methods, thereby providing approximations to the exact marginals. We show how a slightly weakened log-determinant relaxation can be solved even more efficiently by a dual reformulation. When applied to denoising problems in a coupled mixture-of-Gaussian model defined on a binary MRF with cycles, we find that the performance of this log-determinant relaxation is comparable or superior to the widely used sum-product algorithm over a range of experimental conditions.

We consider message passing for probabilistic inference in undirected Gaussian graphical models. We show that for singly connected graphs, message passing yields an algorithm that is equivalent to the application of Gaussian elimination to the solution of a particular system of equations. This relation provides a natural way of extending message passing to arbitrary graphs with loops by first studying the operations required by Gaussian elimination. We thus obtain a finite time convergent algorithm that solves the inference problem exactly and whose complexity grows gradually with the "distance" of the graph to a tree. This algorithm can be implemented in a distributed fashion at nodes through message passing, as for example in sensor networks.

Graphical models provide a powerful formalism for statistical signal processing. Due to their sophis-ticated modeling capabilities, they have found applications in a variety of fields such as computer vision, image processing, and distributed sensor networks. In this paper, we present a general class of algorithms for estimation in Gaussian graphical models with arbitrary structure. These algorithms involve a sequence of inference problems on tractable subgraphs over subsets of variables. This framework includes parallel iterations such as Embedded Trees, serial iterations such as block Gauss-Seidel, and hybrid versions of these iterations. We also discuss a method that uses local memory at each node to overcome temporary communication failures that may arise in distributed sensor network applications. We analyze these algorithms based on the recently developed walk-sum interpretation of Gaussian inference. We describe the walks “computed ” by the algorithms using walk-sum diagrams, and show that arbitrary non-stationary iterations (i.e., based on any sequence of subgraphs) of the algorithms converge in walk-summable models. Since subgraphs can be used in any order, we are free to choose spanning trees and subsets of variables adaptively at each iteration. This leads to efficient methods for optimizing the next iteration step to achieve maximum reduction in error. Simulation results demonstrate that these non-stationary algorithms provide a significant speedup in convergence over traditional one-tree and two-tree iterations.

Abstract—We propose a new iterative, distributed approach for linear minimum mean-square-error (LMMSE) estimation in graphical models with cycles. The embedded subgraphs algorithm (ESA) decomposes a loopy graphical model into a number of linked embedded subgraphs and applies the classical parallel block Jacobi iteration comprising local LMMSE estimation in each subgraph (involving inversion of a small matrix) followed by an information exchange between neighboring nodes and subgraphs. Our primary application is sensor networks, where the model encodes the correlation structure of the sensor measurements, which are assumed to be Gaussian. The resulting LMMSE estimation problem involves a large matrix inverse, which must be solved in-network with distributed computation and minimal intersensor communication. By invoking the theory of asynchronous iterations, we prove that ESA is robust to temporary communication faults such as failing links and sleeping nodes, and enjoys guaranteed convergence under relatively mild conditions. Simulation studies demonstrate that ESA compares favorably with other recently proposed algorithms for distributed estimation. Simulations also indicate that energy consumption for iterative estimation increases substantially as more links fail or nodes sleep. Thus, somewhat surprisingly, sensor network energy conservation strategies such as low-powered transmission and aggressive sleep schedules could actually prove counterproductive. Our results can be replicated using MATLAB code from www.dsp.rice.edu/software. Index Terms—Asynchronous iterations, distributed estimation, graphical models, matrix splitting, sensor networks, Wiener filter. I.

Smoothing approaches to the Simultaneous Localization and Mapping (SLAM) problem in robotics are superior to the more common filtering approaches in being exact, better equipped to deal with non-linearities, and computing the entire robot trajectory. However, while filtering algorithms that perform map updates in constant time exist, no analogous smoothing method is available. We aim to rectify this situation by presenting a smoothingbased solution to SLAM using Loopy Belief Propagation (LBP) that can perform the trajectory and map updates in constant time except when a loop is closed in the environment. The SLAM problem is represented as a Gaussian Markov Random Field (GMRF) over which LBP is performed. We prove that LBP, in this case, is equivalent to Gauss-Seidel relaxation of a linear system. The inability to compute marginal covariances efficiently in a smoothing algorithm has previously been a stumbling block to their widespread use. LBP enables the efficient recovery of the marginal covariances, albeit approximately, of landmarks and poses. While the final covariances are overconfident, the ones obtained from a spanning tree of the GMRF are conservative, making them useful for data association. Experiments in simulation and using real data are presented. 1

Abstract—We present a versatile framework for tractable computation of approximate variances in large-scale Gaussian Markov random field estimation problems. In addition to its efficiency and simplicity, it also provides accuracy guarantees. Our approach relies on the construction of a certain low-rank aliasing matrix with respect to the Markov graph of the model. We first construct this matrix for single-scale models with short-range correlations and then introduce spliced wavelets and propose a construction for the long-range correlation case, and also for multiscale models. We describe the accuracy guarantees that the approach provides and apply the method to a large interpolation problem from oceanography with sparse, irregular, and noisy measurements, and to a gravity inversion problem. Index Terms—Approximate variances, Gaussian Markov random fields, multiscale models, wavelets.

We consider the problem of variance estimation in large-scale Gauss-Markov random field (GMRF) models. While approximate mean estimates can be obtained efficiently for sparse GMRFs of very large size, computing the variances is a challenging problem. We propose a simple rank-reduced method which exploits the graph structure and the correlation length in the model to compute approximate variances with linear complexity in the number of nodes. The method has a separation length parameter trading off complexity versus estimation accuracy. For models with bounded correlation length, we efficiently compute provably accurate variance estimates.