Permutations Unlabeled beyond Sampling Unknown

A recent unlabeled sampling result by Unnikrishnan, Haghighatshoar and Vetterli states that with probability one over iid Gaussian matrices A, any x can be uniquely recovered from an unknown permutation of y = A x as soon as A has at least twice as many rows as columns. We show that this condition on A implies something much stronger: that an unknown vector x can be recovered from measurements y = T A x, when the unknown T belongs to an arbitrary set of invertible, diagonalizable linear transformations T. The set T can be finite or countably infinite. When it is the set of m × m permutation matrices, we have the classical unlabeled sampling problem. We show that for almost all A with at least twice as many rows as columns, all x can be recovered either uniquely, or up to a scale depending on T, and that the condition on the size of A is necessary. Our proof is based on vector space geometry. Specializing to permutations we obtain a simplified proof of the uniqueness result of Unnikrishnan, Haghighatshoar and Vetterli. In this letter we are only concerned with uniqueness; stability and algorithms are left for future work.