Ground Metric Learning

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.

32 pages, 4 figures

References in corpus (1)

• A Survey on Metric Learning for Feature Vectors and Structured Data

Cited by in corpus (15)

• Wasserstein Distributionally Robust Kalman Filtering
• Supervised LogEuclidean Metric Learning for Symmetric Positive Definite Matrices
• Regularized Optimal Transport and the Rot Mover's Distance
• Semi-Supervised Multi-Modal Multi-Instance Multi-Label Deep Network with Optimal Transport
• Ground Metric Learning on Graphs
• Regularized Optimal Transport is Ground Cost Adversarial
• Two-Stage Metric Learning
• Learning Cost Functions for Optimal Transport
• Neural Topic Model via Optimal Transport
• Geometry of Graph Partitions via Optimal Transport
• Making transport more robust and interpretable by moving data through a small number of anchor points
• Learning transport cost from subset correspondence
• Information Geometry Connecting Wasserstein Distance and Kullback-Leibler Divergence via the Entropy-Relaxed Transportation Problem
• Interpolated Discretized Embedding of Single Vectors and Vector Pairs for Classification, Metric Learning and Distance Approximation
• Selective information exchange in collaborative clustering using regularized Optimal Transport