Learning Shared Kernel Models: the Shared Kernel EM algorithm

TL;DR

This paper introduces the shared kernel EM (SKEM) algorithm, a rigorous derivation of a shared kernel mixture model, applied to digit recognition, with comparisons of variants and source code provided.

Contribution

It provides a theoretically rigorous derivation of the shared kernel EM algorithm for Gaussian mixture models, addressing previous shortcomings and applying it to digit recognition.

Findings

SKEM achieves competitive accuracy in digit recognition.

Different feature and mixture component configurations impact performance.

Simplified classifiers decompose joint PDFs for efficiency.

Abstract

Expectation maximisation (EM) is an unsupervised learning method for estimating the parameters of a finite mixture distribution. It works by introducing "hidden" or "latent" variables via Baum's auxiliary function $Q$ that allow the joint data likelihood to be expressed as a product of simple factors. The relevance of EM has increased since the introduction of the variational lower bound (VLB): the VLB differs from Baum's auxiliary function only by the entropy of the PDF of the latent variables $Z$. We first present a rederivation of the standard EM algorithm using data association ideas from the field of multiple target tracking, using $K$-valued scalar data association hypotheses rather than the usual binary indicator vectors. The same method is then applied to a little known but much more general type of supervised EM algorithm for shared kernel models, related to probabilistic radial basis function networks. We address a number of shortcomings in the derivations that have been published previously in this area. In particular, we give theoretically rigorous derivations of (i) the complete data likelihood; (ii) Baum's auxiliary function (the E-step) and (iii) the maximisation (M-step) in the case of Gaussian shared kernel models. The subsequent algorithm, called shared kernel EM (SKEM), is then applied to a digit recognition problem using a novel 7-segment digit representation. Variants of the algorithm that use different numbers of features and different EM algorithm dimensions are compared in terms of mean accuracy and mean IoU. A simplified classifier is proposed that decomposes the joint data PDF as a product of lower order PDFs over non-overlapping subsets of variables. The effect of different numbers of assumed mixture components $K$ is also investigated. High-level source code for the data generation and SKEM algorithm is provided.