Learning Choquet-Integral-Based Metrics for Semisupervised Clustering

We consider an application of fuzzy measures to the problem of metric learning in semisupervised clustering. We investigate the necessary and sufficient conditions on the underlying fuzzy measure that make the discrete Choquet integral suitable for defining a metric. As a byproduct, we can obtain the analogous conditions for the ordered-weighted-averaging (OWA) operators, which constitute a special case. We then generalize these results for power-based Choquet and OWA operators. We show that this metric-learning problem can be formulated as a linear-programming problem and specify the required sets of linear constraints. We present the results of numerical experiments on artificial- and real-world datasets, which illustrate the potential, usefulness, and limitations of this construction.