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Homomorphic sensing of subspace arrangements
Applied and Computational Harmonic Analysis  (IF3.055),  Pub Date : 2021-07-30, DOI: 10.1016/j.acha.2021.06.008
Liangzu Peng, Manolis C. Tsakiris

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. We provide tighter and simpler conditions guaranteeing the unique recovery for the single-subspace case, and we extend the result to subspace arrangements and noisy measurements. We specialize our results to homomorphic sensing examples such as real phase retrieval and unlabeled sensing. In so doing, in a unified way, we obtain conditions guaranteeing 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.