Multiview Sensing With Unknown Permutations: An Optimal Transport Approach

TL;DR

This paper addresses the challenge of recovering signals from measurements with unknown permutations by leveraging optimal transport, introducing a regularization that favors likely permutations and developing a tractable algorithm despite the problem's non-convexity.

Contribution

It presents a novel OT-based framework with regularization for permutation recovery, enabling practical solutions in complex sensing scenarios.

Findings

Regularization improves permutation likelihood estimation.

Relaxation makes the problem computationally tractable.

OT-based approach outperforms existing methods in certain applications.

Abstract

In several applications, including imaging of deformable objects while in motion, simultaneous localization and mapping, and unlabeled sensing, we encounter the problem of recovering a signal that is measured subject to unknown permutations. In this paper we take a fresh look at this problem through the lens of optimal transport (OT). In particular, we recognize that in most practical applications the unknown permutations are not arbitrary but some are more likely to occur than others. We exploit this by introducing a regularization function that promotes the more likely permutations in the solution. We show that, even though the general problem is not convex, an appropriate relaxation of the resulting regularized problem allows us to exploit the well-developed machinery of OT and develop a tractable algorithm.