Compressed sensing with unknown sensor permutation

Abstract

International audienceCompressed sensing is the ability to retrieve a sparse vector from a set of linear measurements. The task gets more difficult when the sensing process is not perfectly known. We address such a problem in the case where the sensors have been permuted, i.e., the order of the measurements is unknown. We propose a branch-and-bound algorithm that converges to the solution. The experimental study shows that our approach always retrieves the unknown permutation, while a simple convex relaxation strategy almost always fails. In terms of its time complexity, we show that the proposed algorithm converges quickly with respect to the combinatorial nature of the problem

Unlabeled sensing: Solving a linear system with unordered measurements

“…By expanding the determinant with respect to the first and the third row of D, it can be checked that det(D) is again quadratic in terms of λ, where the coefficient of λ 2 is given by the determinant ofD whereD given by Eq. (13).…”

“…They propose a branch-and-bound algorithm for solving the problem. The main difference in the framework of that work from ours is the fact that in [13] multiple observations are available under the same permutation matrix Π which, together with the sparsity assumption, simplifies the task of estimating the unknown permutation.…”

“…For example, in [13], a variant of the problem in (1) with repeated observations was studied under a sparsity assumption on x in the context of dictionary learning. The authors assume that the permutation matrix Π remains invariant for multiple observations so that the effective sensing system is of the same form as (1) but with vectors x and y replaced with matrices with T columns each, where T represents the number of observations.…”

Parameter Estimation via Unlabeled Sensing Using Distributed Sensors

“…A variant, where the measurement matrix suffers from errors is also studied, and the corresponding maximum likelihood (ML) estimator can be found efficiently via numerical algorithms [3][4][5][6]. Recently, a parameter estimation problem where the observations are perturbed by an unknown permutation matrix has been studied extensively [7][8][9][10][11]. In [7], a branch and bound algorithm utilizing sparsity information is proposed and numerical results show that the unknown permutation can be correctly retrieved.…”

“…Recently, a parameter estimation problem where the observations are perturbed by an unknown permutation matrix has been studied extensively [7][8][9][10][11]. In [7], a branch and bound algorithm utilizing sparsity information is proposed and numerical results show that the unknown permutation can be correctly retrieved. In the absence of noise, it is shown that as long as the number of measurements is twice as many as that of unknown parameters under random linear sensing strategy, exact recovery is possible [8].…”