Diffusion Methods for Classification with Pairwise Relationships

TL;DR

This paper introduces two convergent algorithms for classification problems with pairwise relationships, leveraging contraction maps, and relates them to message passing, diffusion, and Markov decision processes, with applications in image restoration and stereo depth estimation.

Contribution

The paper presents novel algorithms guaranteed to converge on arbitrary graphs, linking message passing, diffusion, and decision processes, with proven theoretical properties and practical applications.

Findings

Algorithms always converge to a unique fixed point.

Fixed points provide lower bounds on energy and max-marginals.

Successful applications demonstrated in image restoration and stereo depth estimation.

Abstract

We define two algorithms for propagating information in classification problems with pairwise relationships. The algorithms are based on contraction maps and are related to non-linear diffusion and random walks on graphs. The approach is also related to message passing algorithms, including belief propagation and mean field methods. The algorithms we describe are guaranteed to converge on graphs with arbitrary topology. Moreover they always converge to a unique fixed point, independent of initialization. We prove that the fixed points of the algorithms under consideration define lower-bounds on the energy function and the max-marginals of a Markov random field. The theoretical results also illustrate a relationship between message passing algorithms and value iteration for an infinite horizon Markov decision process. We illustrate the practical application of the algorithms under study with numerical experiments in image restoration, stereo depth estimation and binary classification on a grid.