r-local sensing: Improved algorithm and applications

TL;DR

This paper introduces an efficient algorithm for the r-local unlabeled sensing problem, which involves noisy linear systems with block-structured unknown permutations, and demonstrates its effectiveness on synthetic and real data.

Contribution

It proposes a proximal alternating minimization algorithm tailored for r-local permutation constraints, with proven convergence and practical efficiency.

Findings

Algorithm converges to a first order stationary point.

Efficient performance demonstrated on synthetic datasets.

Validated on real datasets with promising results.

Abstract

The unlabeled sensing problem is to solve a noisy linear system of equations under unknown permutation of the measurements. We study a particular case of the problem where the permutations are restricted to be r-local, i.e. the permutation matrix is block diagonal with r x r blocks. Assuming a Gaussian measurement matrix, we argue that the r-local permutation model is more challenging compared to a recent sparse permutation model. We propose a proximal alternating minimization algorithm for the general unlabeled sensing problem that provably converges to a first order stationary point. Applied to the r-local model, we show that the resulting algorithm is efficient. We validate the algorithm on synthetic and real datasets. We also formulate the 1-d unassigned distance geometry problem as an unlabeled sensing problem with a structured measurement matrix.