Low-rank optimization for distance matrix completion

TL;DR

This paper introduces scalable algorithms for low-rank distance matrix completion in high-dimensional settings, capable of recovering missing data entries and estimating the embedding dimension with proven convergence.

Contribution

It proposes two efficient algorithms for low-rank distance matrix completion and a method to determine the embedding space dimension, scalable to high-dimensional data.

Findings

Algorithms scale well to high-dimensional problems

Monotonic convergence to a global solution

Good performance demonstrated on benchmark datasets

Abstract

This paper addresses the problem of low-rank distance matrix completion. This problem amounts to recover the missing entries of a distance matrix when the dimension of the data embedding space is possibly unknown but small compared to the number of considered data points. The focus is on high-dimensional problems. We recast the considered problem into an optimization problem over the set of low-rank positive semidefinite matrices and propose two efficient algorithms for low-rank distance matrix completion. In addition, we propose a strategy to determine the dimension of the embedding space. The resulting algorithms scale to high-dimensional problems and monotonically converge to a global solution of the problem. Finally, numerical experiments illustrate the good performance of the proposed algorithms on benchmarks.