When Does an Ensemble of Matrices with Randomly Scaled Rows Lose Rank?
Abstract
We consider the problem of determining rank loss conditions for a concatenation of full-rank matrices, such that each row of the composing matrices is scaled by a random coefficient. This problem has applications in wireless interference management and recommendation systems. We determine necessary and sufficient conditions for the design of each matrix, such that the random ensemble will almost surely lose rank by a certain amount. The result is proved by converting the problem to determining rank loss conditions for the union of some specific matroids, and then using tools from matroid and graph theories to derive the necessary and sufficient conditions. As an application, we discuss how this result can be applied to the problem of topological interference management, and characterize the linear symmetric degrees of freedom for a class of network topologies.
Keywords
Cite
@article{arxiv.1501.07544,
title = {When Does an Ensemble of Matrices with Randomly Scaled Rows Lose Rank?},
author = {Navid Naderializadeh and Aly El Gamal and A. Salman Avestimehr},
journal= {arXiv preprint arXiv:1501.07544},
year = {2015}
}
Comments
submitted to IEEE Transactions on Information Theory; shorter version to appear at IEEE International Symposium on Information Theory (ISIT 2015)