Permutation Recovery from Multiple Measurement Vectors in Unlabeled Sensing

Abstract

In Unlabeled Sensing, one observes a set of linear measurements of an underlying signal with incomplete or missing information about their ordering, which can be modeled in terms of an unknown permutation. In this paper, we study the case of multiple noisy measurement vectors (MMVs) resulting from a common permutation and investigate to what extent the number of MMVs m facilitates permutation recovery by borrowing strength. We provide an affirmative answer for an oracle setting in which the matrix of signals is known by establishing matching upper and lower bounds on the required Signal-to-Noise Ratio (SNR), which – as distinguished from the case of a single measurement vector – involves a dependence on the stable rank of the matrix of signals. Specifically, a larger stable rank significantly reduces the required average SNR which can drop from nΩ(1) for m = 1 to Ω(log n) for m = Ω(log n), where n denotes the number of measurements per MMV. Numerical results are well-aligned with our theoretical findings.