Dirichlet Aggregation: Unsup ervised Learning towards an Optimal Metric for Prop ortional Data

Dirichlet Aggregation: Unsup ervised Learning towards an Optimal Metric for Prop ortional Data
Hua-Yan Wang - State Key Laboratory of Machine Perception, Peking University, China Hongbin Zha - State Key Laboratory of Machine Perception, Peking University, China Hong Qin - Department of Computer Science, State University of New York at Stony Brook, USA
Proportional data (normalized histograms) have been frequently occurring in various areas, and they could be mathematically abstracted as points residing in a geometric simplex. A proper distance metric on this simplex is of importance in many applications including classification and information retrieval. In this paper, we develop a novel framework to learn an optimal metric on the simplex. Ma jor features of our approach include: 1) its flexibility to handle correlations among bins/dimensions; 2) widespread applicability without being limited to ad hoc backgrounds; and 3) a "real" global solution in contrast to existing traditional local approaches. The technical essence of our approach is to fit a parametric distribution to the observed empirical data in the simplex. The distribution is parameterized by affinities between simplex vertices, which is learned via maximizing likelihood of observed data. Then, these affinities induce a metric on the simplex, defined as the earth mover's distance equipped with ground distances derived from simplex vertex affinities.

Hua-Yan Wang - State Key Laboratory of Machine Perception, Peking University, China
Hongbin Zha - State Key Laboratory of Machine Perception, Peking University, China
Hong Qin - Department of Computer Science, State University of New York at Stony Brook, USA

Proportional data (normalized histograms) have been frequently occurring in various areas, and they could be mathematically abstracted as points residing in a geometric simplex. A proper distance metric on this simplex is of importance in many applications including classification and information retrieval. In this paper, we develop a novel framework to learn an optimal metric on the simplex. Ma jor features of our approach include: 1) its flexibility to handle correlations among bins/dimensions; 2) widespread applicability without being limited to ad hoc backgrounds; and 3) a "real" global solution in contrast to existing traditional local approaches. The technical essence of our approach is to fit a parametric distribution to the observed empirical data in the simplex. The distribution is parameterized by affinities between simplex vertices, which is learned via maximizing likelihood of observed data. Then, these affinities induce a metric on the simplex, defined as the earth mover's distance equipped with ground distances derived from simplex vertex affinities.