Support Measure Data Description

Jorge Guevara (1) , S Ephane Canu (2) , R Hirata (3)

1 VISION - Computer vision research group
2 INSA Rouen Normandie - Institut national des sciences appliquées Rouen Normandie
3 USP - Universidade de São Paulo = University of São Paulo

—We address the problem of learning a data description model for a dataset whose elements or observations are itself a set of points in R D . Modeling each observation as a probability measure, we describe such dataset by computing a minimum volume set for the probability measures, as means of a minimum enclosing ball of the representer functions of the probability measures in a Reproducing Kernel Hilbert Space (RKHS). The advantage is that we do not consider any particular form for the probability measures, instead, we use the embedding of such measures into a RKHS given by a positive definite kernel on probability measures. As a result, the data description model is a function that only depends on some probability measures: the support measures. We formulated three support measure data description models for such datasets: the optimization problem for the first one is a chance constrained program; the second is a direct extension of the support vector data description method to the case of probability measures; the third is the same as the second one, but defined for stationary kernels and scaling on data. We validate our method in the challenging task of group anomaly detection, with artificial and real datasets.

Mots clés: mean map; group anomaly detection; Index Terms—Kernel on distributions; One-class classification; support vector data description; embedding of probability measures

Domaines: Apprentissage [cs.LG]

Jorge Guevara, S Ephane Canu, R Hirata. Support Measure Data Description. 2014. ⟨hal-01015718v3⟩