Convolutional Factor Graphs as Probabilistic Models

TL;DR

This paper introduces convolutional factor graphs (CFGs) as a new class of probabilistic models that effectively represent linear combinations of independent latent variables, with a duality property enabling efficient inference.

Contribution

It defines CFGs as a novel probabilistic graphical model class inspired by error control coding, highlighting their structure, properties, and duality with multiplicative factor graphs.

Findings

CFGs model probability functions involving linear transformations of independent latent variables.

Fourier transform duality enables inference on CFGs to be performed via their dual MFGs.

Demonstrations with Gaussian and independent factor models illustrate CFGs' applicability.

Abstract

Based on a recent development in the area of error control coding, we introduce the notion of convolutional factor graphs (CFGs) as a new class of probabilistic graphical models. In this context, the conventional factor graphs are referred to as multiplicative factor graphs (MFGs). This paper shows that CFGs are natural models for probability functions when summation of independent latent random variables is involved. In particular, CFGs capture a large class of linear models, where the linearity is in the sense that the observed variables are obtained as a linear ransformation of the latent variables taking arbitrary distributions. We use Gaussian models and independent factor models as examples to emonstrate the use of CFGs. The requirement of a linear transformation between latent variables (with certain independence restriction) and the bserved variables, to an extent, limits the modelling flexibility of CFGs. This structural restriction however provides a powerful analytic tool to the framework of CFGs; that is, upon taking the Fourier transform of the function represented by the CFG, the resulting function is represented by a FG with identical structure. This Fourier transform duality allows inference problems on a CFG to be solved on the corresponding dual MFG.