[Paper Review] Negative Tree Reweighted Belief Propagation

Qiang Liu, Alexander Ihler|arXiv (Cornell University)|Mar 15, 2012

Error Correcting Code Techniques6 references4 citations

TL;DR

This paper introduces Negative Tree Reweighted Belief Propagation (NTRBP), a novel lower-bound approximation for the log partition function in Markov random fields using linear combinations of spanning trees with negative weights. By reversing Jensen's inequality, NTRBP provides tighter lower bounds than mean field methods and generalizes structured mean field approximations, offering a non-convex optimization framework for improved variational inference in graphical models.

ABSTRACT

We introduce a new class of lower bounds on the log partition function of a Markov random field which makes use of a reversed Jensen's inequality. In particular, our method approximates the intractable distribution using a linear combination of spanning trees with negative weights. This technique is a lower-bound counterpart to the tree-reweighted belief propagation algorithm, which uses a convex combination of spanning trees with positive weights to provide corresponding upper bounds. We develop algorithms to optimize and tighten the lower bounds over the non-convex set of valid parameter values. Our algorithm generalizes mean field approaches (including naive and structured mean field approximations), which it includes as a limiting case.

Motivation & Objective

• To develop a tighter lower bound on the log partition function of Markov random fields, which is typically intractable to compute exactly.
• To overcome limitations of existing upper-bound methods like tree-reweighted belief propagation by introducing a dual lower-bound approach.
• To generalize mean field approximations, including naive and structured variants, by formulating them as limiting cases of the proposed framework.
• To optimize the lower bound over the non-convex set of valid parameter values, enabling improved variational inference.
• To provide a new variational inference technique that complements existing upper-bound methods and enhances approximation accuracy.

Proposed method

• Proposes a lower bound on the log partition function using a reversed Jensen's inequality applied to a linear combination of spanning trees with negative weights.
• Constructs a variational approximation by combining spanning trees with negative coefficients, forming a non-convex optimization problem over valid parameter sets.
• Derives an objective function that maximizes the lower bound, leveraging the structure of spanning trees to model dependencies in the graphical model.
• Develops iterative algorithms to optimize the lower bound, extending belief propagation principles to negative-weight tree combinations.
• Establishes a connection between the proposed method and mean field approximations, showing that mean field arises as a limiting case when tree weights approach zero.
• Uses a non-convex optimization framework to tighten the lower bound, enabling better approximation of the true partition function.

Experimental results

Research questions

• RQ1 Can a reversed Jensen's inequality be used to derive a tighter lower bound on the log partition function in Markov random fields?
• RQ2 How can spanning trees with negative weights be used to construct a valid variational approximation for intractable graphical models?
• RQ3 In what way does the proposed method generalize or subsume existing mean field approaches?
• RQ4 Can the non-convex optimization of negative-weight tree combinations yield better variational bounds than standard mean field or tree-reweighted methods?
• RQ5 What is the relationship between the proposed lower-bound method and the upper-bound tree-reweighted belief propagation algorithm?

Key findings

• The proposed method provides a valid lower bound on the log partition function by exploiting a reversed Jensen's inequality, enabling tighter approximations than standard mean field methods.
• The framework generalizes both naive and structured mean field approximations, which emerge as limiting cases when tree weights approach zero.
• The method achieves improved variational bounds by optimizing over a non-convex set of negative-weight spanning tree combinations, offering a dual to the convex upper-bound approach of tree-reweighted belief propagation.
• Empirical results demonstrate that the lower bounds are tighter than those from mean field approximations, particularly in models with strong dependencies.
• The algorithm successfully extends belief propagation principles to negative-weight tree combinations, enabling new inference capabilities in graphical models.
• The approach provides a complementary alternative to upper-bound methods, enhancing the overall toolkit for variational inference in Markov random fields.