1,426 research outputs found
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
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