Combined Belief Propagation-Mean Field Message Passing Algorithm for Dirichlet Process Mixtures

Robots working in human environments often encounter a wide range of articulated objects, such as tools, cabinets, and other jointed objects. Such articulated objects can take an infinite number of possible poses, as a point in a potentially high-dimensional continuous space. A robot must perceive this continuous pose to manipulate the object to a desired pose. This problem of perception and manipulation of articulated objects remains a challenge due to its high dimensionality and multimodal uncertainty. Here, we describe a factored approach to estimate the poses of articulated objects using an efficient approach to nonparametric belief propagation. We consider inputs as geometrical models with articulation constraints and observed RGBD (red, green, blue, and depth) sensor data. The described framework produces object-part pose beliefs iteratively. The problem is formulated as a pairwise Markov random field (MRF), where each hidden node (continuous pose variable) is an observed object-part’s pose and the edges denote the articulation constraints between the parts. We describe articulated pose estimation by a “pull” message passing algorithm for nonparametric belief propagation (PMPNBP) and evaluate its convergence properties over scenes with articulated objects. Robot experiments are provided to demonstrate the necessity of maintaining beliefs to perform goal-driven manipulation tasks.

Download Full-text

A Spectral Approach to Analysing Belief Propagation for 3-Colouring

Combinatorics Probability Computing ◽

10.1017/s096354830900981x ◽

2009 ◽

Vol 18 (6) ◽

pp. 881-912 ◽

Cited By ~ 14

Author(s):

AMIN COJA-OGHLAN ◽

ELCHANAN MOSSEL ◽

DAN VILENCHIK

Keyword(s):

Graphical Models ◽

Message Passing ◽

Graphical Model ◽

Belief Propagation ◽

Graph Colouring ◽

Rigorous Result ◽

Theoretical Understanding ◽

Message Passing Algorithm ◽

Survey Propagation ◽

Graphical Structure

Belief propagation (BP) is a message-passing algorithm that computes the exact marginal distributions at every vertex of a graphical model without cycles. While BP is designed to work correctly on trees, it is routinely applied to general graphical models that may contain cycles, in which case neither convergence, nor correctness in the case of convergence is guaranteed. Nonetheless, BP has gained popularity as it seems to remain effective in many cases of interest, even when the underlying graph is ‘far’ from being a tree. However, the theoretical understanding of BP (and its new relative survey propagation) when applied to CSPs is poor.Contributing to the rigorous understanding of BP, in this paper we relate the convergence of BP to spectral properties of the graph. This encompasses a result for random graphs with a ‘planted’ solution; thus, we obtain the first rigorous result on BP for graph colouring in the case of a complex graphical structure (as opposed to trees). In particular, the analysis shows how belief propagation breaks the symmetry between the 3! possible permutations of the colour classes.

Download Full-text

Recovering a hidden community beyond the Kesten–Stigum threshold in O(|E|log*|V|) time

Journal of Applied Probability ◽

10.1017/jpr.2018.22 ◽

2018 ◽

Vol 55 (2) ◽

pp. 325-352 ◽

Cited By ~ 3

Author(s):

Bruce Hajek ◽

Yihong Wu ◽

Jiaming Xu

Keyword(s):

Message Passing ◽

Belief Propagation ◽

Linear Time ◽

Signal To Noise Ratio ◽

Bp Algorithm ◽

Local Algorithm ◽

Message Passing Algorithm ◽

Edge Probability ◽

Effective Signal ◽

Total Time Complexity

Abstract Community detection is considered for a stochastic block model graph of n vertices, with K vertices in the planted community, edge probability p for pairs of vertices both in the community, and edge probability q for other pairs of vertices. The main focus of the paper is on weak recovery of the community based on the graph G, with o(K) misclassified vertices on average, in the sublinear regime n1-o(1) ≤ K ≤ o(n). A critical parameter is the effective signal-to-noise ratio λ = K2(p - q)2 / ((n - K)q), with λ = 1 corresponding to the Kesten–Stigum threshold. We show that a belief propagation (BP) algorithm achieves weak recovery if λ > 1 / e, beyond the Kesten–Stigum threshold by a factor of 1 / e. The BP algorithm only needs to run for log*n + O(1) iterations, with the total time complexity O(|E|log*n), where log*n is the iterated logarithm of n. Conversely, if λ ≤ 1 / e, no local algorithm can asymptotically outperform trivial random guessing. Furthermore, a linear message-passing algorithm that corresponds to applying a power iteration to the nonbacktracking matrix of the graph is shown to attain weak recovery if and only if λ > 1. In addition, the BP algorithm can be combined with a linear-time voting procedure to achieve the information limit of exact recovery (correctly classify all vertices with high probability) for all K ≥ (n / logn) (ρBP + o(1)), where ρBP is a function of p / q.

Download Full-text

Properties of Bethe Free Energies and Message Passing in Gaussian Models

Journal of Artificial Intelligence Research ◽

10.1613/jair.3195 ◽

2011 ◽

Vol 41 ◽

pp. 1-24 ◽

Cited By ~ 2

Author(s):

B. Cseke ◽

T. Heskes

Keyword(s):

Free Energy ◽

Message Passing ◽

Probabilistic Models ◽

Mean Field ◽

Necessary Condition ◽

Free Energies ◽

Gaussian Models ◽

Message Passing Algorithm ◽

Bethe Free Energy ◽

The Moment

We address the problem of computing approximate marginals in Gaussian probabilistic models by using mean field and fractional Bethe approximations. We define the Gaussian fractional Bethe free energy in terms of the moment parameters of the approximate marginals, derive a lower and an upper bound on the fractional Bethe free energy and establish a necessary condition for the lower bound to be bounded from below. It turns out that the condition is identical to the pairwise normalizability condition, which is known to be a sufficient condition for the convergence of the message passing algorithm. We show that stable fixed points of the Gaussian message passing algorithm are local minima of the Gaussian Bethe free energy. By a counterexample, we disprove the conjecture stating that the unboundedness of the free energy implies the divergence of the message passing algorithm.

Download Full-text

Hybrid Message-Passing Algorithm and Architecture for Decoding Cyclic Non-binary LDPC Codes

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences ◽

10.1587/transfun.e96.a.2652 ◽

2013 ◽

Vol E96.A (12) ◽

pp. 2652-2659

Author(s):

Yichao LU ◽

Gang HE ◽

Guifen TIAN ◽

Satoshi GOTO

Keyword(s):

Message Passing ◽

Ldpc Codes ◽

Message Passing Algorithm

Download Full-text

Modules or Mean-Fields?

Entropy ◽

10.3390/e22050552 ◽

2020 ◽

Vol 22 (5) ◽

pp. 552 ◽

Cited By ~ 2

Author(s):

Thomas Parr ◽

Noor Sajid ◽

Karl J. Friston

Keyword(s):

Steady State ◽

Message Passing ◽

Mean Field Theory ◽

Functional Anatomy ◽

Mean Field ◽

State Density ◽

Stochastic Dynamical Systems ◽

Conditional Independency ◽

The Brain ◽

Non Equilibrium

The segregation of neural processing into distinct streams has been interpreted by some as evidence in favour of a modular view of brain function. This implies a set of specialised ‘modules’, each of which performs a specific kind of computation in isolation of other brain systems, before sharing the result of this operation with other modules. In light of a modern understanding of stochastic non-equilibrium systems, like the brain, a simpler and more parsimonious explanation presents itself. Formulating the evolution of a non-equilibrium steady state system in terms of its density dynamics reveals that such systems appear on average to perform a gradient ascent on their steady state density. If this steady state implies a sufficiently sparse conditional independency structure, this endorses a mean-field dynamical formulation. This decomposes the density over all states in a system into the product of marginal probabilities for those states. This factorisation lends the system a modular appearance, in the sense that we can interpret the dynamics of each factor independently. However, the argument here is that it is factorisation, as opposed to modularisation, that gives rise to the functional anatomy of the brain or, indeed, any sentient system. In the following, we briefly overview mean-field theory and its applications to stochastic dynamical systems. We then unpack the consequences of this factorisation through simple numerical simulations and highlight the implications for neuronal message passing and the computational architecture of sentience.

Download Full-text