Homomorphic Sensing of Subspace Arrangements
This work addresses recovery problems in linear algebra and signal processing, offering incremental improvements to the homomorphic sensing framework with applications in data with missing or disordered values.
The paper tackles the problem of uniquely recovering points from linear subspaces under linear maps, providing tighter conditions for single subspaces, extending to subspace arrangements, and proving local stability under noise. It applies these results to examples like real phase retrieval and unlabeled sensing, deriving unified conditions and novel ones for sparse and unsigned versions.
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.