Ground Metric Learning

TL;DR

This paper introduces a novel method called ground metric learning that automatically learns the optimal ground metric for transportation distances from labeled data, enhancing their practical applicability in machine learning tasks.

Contribution

It formulates the problem of learning the ground metric as a convex optimization task and proposes algorithms to solve it, enabling data-driven selection of the metric.

Findings

Effective in binary image classification tasks

Outperforms fixed metric approaches

Demonstrates practical viability on Caltech-256 dataset

Abstract

Transportation distances have been used for more than a decade now in machine learning to compare histograms of features. They have one parameter: the ground metric, which can be any metric between the features themselves. As is the case for all parameterized distances, transportation distances can only prove useful in practice when this parameter is carefully chosen. To date, the only option available to practitioners to set the ground metric parameter was to rely on a priori knowledge of the features, which limited considerably the scope of application of transportation distances. We propose to lift this limitation and consider instead algorithms that can learn the ground metric using only a training set of labeled histograms. We call this approach ground metric learning. We formulate the problem of learning the ground metric as the minimization of the difference of two polyhedral convex functions over a convex set of distance matrices. We follow the presentation of our algorithms with promising experimental results on binary classification tasks using GIST descriptors of images taken in the Caltech-256 set.