The Metric Nearness Problems with Applications

Abstract

Many practical applications in machine learning require pairwise distances among a set of objects. It is often desirable that these distance measurements satisfy the properties of a metric, especially the triangle inequality. Applications that could benefit from the metric property include data clustering and metric-based indexing of databases. In this paper, we present the metric nearness problem: Given a dissimilarity matrix, find the “nearest ” matrix of distances that satisfy the triangle inequalities. A weight matrix in the formulation captures the confidence in individual dissimilarity measures, including the case of altogether missing distances. For an important class of nearness measures, the problem can be attacked with convex optimization techniques. A pleasing aspect of this formulation is that we can compute globally optimal solutions. Experiments on some sample dissimilarity matrices are presented, including some from biology.