Unlabeled Sensing With Random Linear Measurements

Abstract

We study the problem of solving a linear sensing system when the observations are unlabeled. Specifically we seek a solution to a linear system of equations ${\mathbf{y}}= {\mathbf{A}} {\mathbf{x}} $ when the order of the observations in the vector ${\mathbf{y}}$ is unknown. Focusing on the setting in which ${\mathbf{A}}$ is a random matrix with i.i.d. entries, we show that if the sensing matrix ${\mathbf{A}}$ admits an oversampling ratio of 2 or higher, then, with probability 1, it is possible to recover ${\mathbf{x}}$ exactly without the knowledge of the order of the observations in ${\mathbf{y}}$ . Furthermore, if ${\mathbf{x}}$ is of dimension $K$ , then any $2K$ entries of ${\mathbf{y}}$ are sufficient to recover ${\mathbf{x}}$ . This result implies the existence of deterministic unlabeled sensing matrices with an oversampling factor of 2 that admit perfect reconstruction. The result is universal in that conditioned on the realization of matrix ${\mathbf{A}}$ , recovery is guaranteed for all possible choices of ${\mathbf{x}}$ . While the proof is constructive, it uses a combinatorial algorithm which is not practical, leaving the question of complexity open. We also analyze a noisy version of the problem and show that local stability is guaranteed by the solution. In particular, for every ${\mathbf{x}}$ , the recovery error tends to zero as the signal-to-noise ratio tends to infinity. The question of universal stability is unclear. In addition, we obtain a converse of the result in the noiseless case: If the number of observations in ${\mathbf{y}}$ is less than $2K$ , then with probability 1, universal recovery fails, i.e., with probability 1, there exist distinct choices of ${\mathbf{x}}$ which lead to the same unordered list of observations in ${\mathbf{y}}$ . We also present extensions of the result of the noiseless case to special cases with non-i.i.d. entries in ${\mathbf{A}}$ , and to a different setting in which the labels of a portion of the observations ${\mathbf{y}}$ are known. In terms of applications, the unlabeled sensing problem is related to data association problems encountered in different domains including robotics where it is appears in a method called “simultaneous localization and mapping”, multi-target tracking applications, and in sampling signals in the presence of jitter.