Ladder Matrix Recovery from Permutations

We give unique recovery guarantees for matrices of bounded rank that have undergone permutations of their entries. We even do this for a more general matrix structure that we call ladder matrices. We use methods and results of commutative algebra and algebraic geometry, for which we include a preparation as needed for a general audience.Comment: 14 double-column page

Results on the algebraic matroid of the determinantal variety

We present a class of base sets of the algebraic matroid of the determinantal variety, which we also conjecture they characterize the matroid. This conjecture is then reduced to a purely combinatorial statement. Our technique consists of interpreting matrix completion from a point of view of linear sections on the Grassmannian and invoking a class of local coordinates described by Sturmfels $\&$ Zelevinsky.Comment: 11 pages, reduced the problem to a purely combinatorial conjectur

Homomorphic Sensing of Subspace Arrangements

Homomorphic sensing is a recent algebraic-geometric framework that studies the unique recovery of points in a linear subspace from their images under a given collection of linear maps. It has been successful in interpreting such a recovery in the case of permutations composed by coordinate projections, an important instance in applications known as unlabeled sensing, which models data that are out of order and have missing values. In this paper, we provide tighter and simpler conditions that guarantee the unique recovery for the single-subspace case, extend the result to the case of a subspace arrangement, and show that the unique recovery in a single subspace is locally stable under noise. We specialize our results to several examples of homomorphic sensing such as real phase retrieval and unlabeled sensing. In so doing, in a unified way, we obtain conditions that guarantee the unique recovery for those examples, typically known via diverse techniques in the literature, as well as novel conditions for sparse and unsigned versions of unlabeled sensing. Similarly, our noise result also implies that the unique recovery in unlabeled sensing is locally stable.Comment: 18 page