Matrix Recovery from Permutations: An Algebraic Geometry Approach

We prove that a generic matrix of bounded rank is uniquely recoverable —up to a permutation of its rows and columns— from an arbitrary permutation of its entries. This can be viewed as an extension of the existing unlabeled sensing theory, from linear spaces, to spaces of matrices of bounded rank. Our result is derived using machinery from commutative algebra and algebraic geometry, developed in a self-contained fashion.