If it ain't broke, don't fix it: Sparse metric repair

TL;DR

This paper introduces three combinatorial algorithms for minimally adjusting noisy distance data to satisfy metric properties, ensuring data integrity with minimal alterations and improved computational efficiency.

Contribution

It presents novel algorithms for sparse metric repair that guarantee minimal changes and efficient computation, advancing data correction methods in metric-based applications.

Findings

Algorithms guarantee the sparsest solution in one setting.

All algorithms effectively repair the metric with minimal data alterations.

Running time is proportional to the cube of data points, reducible with prior info.

Abstract

Many modern data-intensive computational problems either require, or benefit from distance or similarity data that adhere to a metric. The algorithms run faster or have better performance guarantees. Unfortunately, in real applications, the data are messy and values are noisy. The distances between the data points are far from satisfying a metric. Indeed, there are a number of different algorithms for finding the closest set of distances to the given ones that also satisfy a metric (sometimes with the extra condition of being Euclidean). These algorithms can have unintended consequences, they can change a large number of the original data points, and alter many other features of the data. The goal of sparse metric repair is to make as few changes as possible to the original data set or underlying distances so as to ensure the resulting distances satisfy the properties of a metric. In other words, we seek to minimize the sparsity (or the $\ell_0$ "norm") of the changes we make to the distances subject to the new distances satisfying a metric. We give three different combinatorial algorithms to repair a metric sparsely. In one setting the algorithm is guaranteed to return the sparsest solution and in the other settings, the algorithms repair the metric. Without prior information, the algorithms run in time proportional to the cube of the number of input data points and, with prior information we can reduce the running time considerably.