Regularization-based learning of the Choquet integral

A number of data-driven fuzzy measure (FM) learning techniques have been put forth for the fuzzy integral (FI). Examples include quadratic programming, Gibbs sampling, gradient descent, reward and punishment and evolutionary optimization. However, most approaches focus solely on the minimization of the sum of squared error (SSE). Limited attention has been placed on characterizing and subsequently minimizing model (i.e., FM) complexity. Furthermore, the vast majority of learning techniques are highly susceptible to over-fitting and noise. Herein, we explore a regularization approach to learning the FM for the Choquet FI. We investigate the mathematical motivation for such an approach, its applicability and impact on different types of FMs, and its desirable properties for quadratic programming (QP) based optimization. We show that L regularization has a distinct meaning for measure learning and aggregation operators. Experiments are performed and validated with respect to the Shapley index. Specifically, we show that it is possible to reduce the effect of overfitting, we can identify higher quality measures and, if desired, force the learning of fewer numbers of sources.