Learning riemannian metrics

Authors:
Guy Lebanon
Affiliations:
School of Computer Science, Carnegie Mellon University, Pittsburgh, PA
Venue:
UAI'03 Proceedings of the Nineteenth conference on Uncertainty in Artificial Intelligence
Year:
2002

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Abstract

We consider the problem of learning a Riemannian metric associated with a given differentiable manifold and a set of points. Our approach to the problem involves choosing a metric from a parametric family that is based on maximizing the inverse volume of a given dataset of points. From a statistical perspective, it is related to maximum likelihood under a model that assigns probabilities inversely proportional to the Riemannian volume element. We discuss in detail learninga metric on the multinomial simplex where the metric candidates are pull-back metrics of the Fisher information under a continuous group of transformations. When applied to documents, the resulting geodesics resemble, but outperform, the TFIDF cosine similarity measure in classification.