Alternating minimization algorithm with initialization analysis for r-local and k-sparse unlabeled sensing

TL;DR

This paper introduces an alternating minimization algorithm with a new initialization strategy for solving unlabeled sensing problems involving permutation models, demonstrating fast, robust, and superior performance on synthetic and real data.

Contribution

The paper presents a novel AltMin algorithm with initialization bounds for r-local and k-sparse permutation models in unlabeled sensing, improving robustness and efficiency.

Findings

Algorithm is fast and applicable to multiple permutation models.

Initialization bounds depend on number of blocks and shuffles.

Outperforms baseline methods on real datasets.

Abstract

Unlabeled sensing is a linear inverse problem with permuted measurements. We propose an alternating minimization (AltMin) algorithm with a suitable initialization for two widely considered permutation models: partially shuffled/$k$-sparse permutations and $r$-local/block diagonal permutations. Key to the performance of the AltMin algorithm is the initialization. For the exact unlabeled sensing problem, assuming either a Gaussian measurement matrix or a sub-Gaussian signal, we bound the initialization error in terms of the number of blocks $s$ and the number of shuffles $k$. Experimental results show that our algorithm is fast, applicable to both permutation models, and robust to choice of measurement matrix. We also test our algorithm on several real datasets for the linked linear regression problem and show superior performance compared to baseline methods.