Permutation Recovery from Multiple Measurement Vectors in Unlabeled
Sensing
release_7u4dtywmkbaixassg6oucit3mi
by
Hang Zhang, Martin Slawski, Ping Li
2019
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. Previous work on the
case of a single noisy measurement vector has exposed two main challenges: 1) a
high requirement concerning the signal-to-noise ratio (snr), i.e.,
approximately of the order of n^5, and 2) a massive computational burden in
light of NP-hardness in general. 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". The above two challenges have at
least partially been resolved within our work. First, we show that a large
stable rank of the signal significantly reduces the required snr which can drop
from a polynomial in n for m = 1 to a constant for m = Ω(log n),
where m denotes the number of MMVs and n denotes the number of measurements
per MV. This bound is shown to be sharp and is associated with a phase
transition phenomenon. Second, we propose computational schemes for recovering
the unknown permutation in practice. For the "oracle case" with the known
signal, the maximum likelihood (ML) estimator reduces to a linear assignment
problem whose global optimum can be obtained efficiently. For the case in which
both the signal and permutation are unknown, the problem is reformulated as a
bi-convex optimization problem with an auxiliary variable, which can be solved
by the Alternating Direction Method of Multipliers (ADMM). Numerical
experiments based on the proposed computational schemes confirm the tightness
of our theoretical analysis.
In text/plain
format
Archived Files and Locations
application/pdf 567.3 kB
file_2zwv5hzuxzdjxnsfl4ghns5d4q
|
arxiv.org (repository) web.archive.org (webarchive) |
1909.02496v1
access all versions, variants, and formats of this works (eg, pre-prints)