Sample-Efficient Geometry Reconstruction from Euclidean Distances using Non-Convex Optimization

Ipsita Ghosh, Abiy Tasissa, Christian Kümmerle

Keywords: Euclidean distance geometry, non-convex optimization, iteratively reweighted least squares, low-rank, data efficiency, convergence guarnatees, restricted isometry property, dual basis

Abstract:

The problem of finding suitable point embedding or geometric configurations given only Euclidean distance information of point pairs arises both as a core task and as a sub-problem in a variety of machine learning applications. In this paper, we aim to solve this problem given a minimal number of distance samples.
To this end, we leverage continuous and non-convex rank minimization formulations of the problem and establish a local convergence
guarantee for a variant of iteratively reweighted least squares (IRLS), which applies if a minimal random set of observed distances is provided.
As a technical tool, we establish a restricted isometry property (RIP) restricted to a tangent space of the manifold of symmetric rank-$r$ matrices given random Euclidean distance measurements, which might be of independent interest for the analysis of other non-convex approaches. Furthermore, we assess data efficiency, scalability and generalizability of different reconstruction algorithms through numerical experiments with simulated data as well as real-world data, demonstrating the proposed algorithm's ability to identify the underlying geometry from fewer distance samples compared to the state-of-the-art.

The Matlab code can be found at \href{https://github.com/ipsita-ghosh-1/EDG-IRLS}{github\_SEGRED}

Primary Area: Optimization (convex and non-convex, discrete, stochastic, robust)

Submission Number: 21313