Title :
Unlabeled sensing: Solving a linear system with unordered measurements
Author :
Jayakrishnan Unnikrishnan;Saeid Haghighatshoar;Martin Vetterli
Author_Institution :
General Electric Global Research, Niskayuna, NY, USA
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 y = Ax when the order of the observations in the vector y is unknown. Focusing on the setting in which A is a random matrix with i.i.d. entries, we show that if the sensing matrix A admits an oversampling ratio of 2 or higher, then with probability one it is possible to recover x exactly without the knowledge of the order of the observations in y. Furthermore, if x is of dimension K, then any 2K entries of y are sufficient to recover x. This result implies the existence of deterministic unlabeled sensing matrices with an oversampling factor of 2 that admit perfect reconstruction. While the proof is constructive, it uses a combinatorial algorithm which is not practical, leaving the question of complexity open. In terms of applications, the unlabeled sensing problem is related to a popular method in robotics called simultaneous location and mapping (SLAM).
Keywords :
"Robot sensing systems","Probability distribution","Compressed sensing","Mathematical model","Linear systems","Random variables"
Conference_Titel :
Communication, Control, and Computing (Allerton), 2015 53rd Annual Allerton Conference on
DOI :
10.1109/ALLERTON.2015.7447086