Sparsity Promotion Models for the Choquet Integral

In this paper, we present a novel algorithm for learning fuzzy measures for Choquet integration. There are two novel aspects of the algorithm: it seeks to explicitly reduce the number of nonzero parameters in the measure to eliminate noninformative or useless information sources and it uses a Bayesian model for parameter estimation which has not been previously applied to the fuzzy measure learning problem. The method uses a hierarchical model that implements a sparsity promotion algorithm through a Gibbs sampler. This approach builds on the methods proposed by Figueiredo et al which uses expectation maximization (EM) to maximize the least absolute shrinkage and selection operator (LASSO) criterion under a distribution that promotes sparsity. Additional constraints are needed to satisfy the requirements of fuzzy measures. Figueiredo's algorithm does not have a mechanism for imposing these constraints. The constraints are imposed by sequentially exploring the lattice tree of the power set and requiring that each fuzzy measure value assigned to a set lies in the domain of a truncated Gaussian determined by the fuzzy measures of supersets of the set under consideration