Unlabelled Sensing with Priors: Algorithm and Bounds

TL;DR

This paper introduces a new estimator for unlabelled sensing with sparse permutations and known correspondences, providing theoretical error bounds and demonstrating improved reconstruction accuracy in experiments and practical motion estimation.

Contribution

It proposes a novel estimator for unlabelled sensing with partial known correspondences, along with theoretical error bounds and empirical validation showing superior performance.

Findings

The estimator achieves lower reconstruction error with known correspondences.

It outperforms classical robust regression by up to 20% in high permutation regimes.

Practical application demonstrated in non-rigid motion estimation with manual point annotations.

Abstract

In this study, we consider a variant of unlabelled sensing where the measurements are sparsely permuted, and additionally, a few correspondences are known. We present an estimator to solve for the unknown vector. We derive a theoretical upper bound on the $\ell_2$ reconstruction error of the unknown vector. Through numerical experiments, we demonstrate that the additional known correspondences result in a significant improvement in the reconstruction error. Additionally, we compare our estimator with the classical robust regression estimator and we find that our method outperforms it on the normalized reconstruction error metric by up to $20\%$ in the high permutation regimes $(>30\%)$. Lastly, we showcase the practical utility of our framework on a non-rigid motion estimation problem. We show that using a few manually annotated points along point pairs with the key-point (SIFT-based) descriptor pairs with unknown or incorrectly known correspondences can improve motion estimation.