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تسجيل مستخدم جديدHomomorphic Sensing of Subspace Arrangements
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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 unsign
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A recent line of research termed unlabeled sensing and shuffled linear regression has been exploring under great generality the recovery of signals from subsampled and permuted measurements; a challenging problem in diverse fields of data science and
machine learning. In this paper we introduce an abstraction of this problem which we call homomorphic sensing. Given a linear subspace and a finite set of linear transformations we develop an algebraic theory which establishes conditions guaranteeing that points in the subspace are uniquely determined from their homomorphic image under some transformation in the set. As a special case, we recover known conditions for unlabeled sensing, as well as new results and extensions. On the algorithmic level we exhibit two dynamic programming based algorithms, which to the best of our knowledge are the first working solutions for the unlabeled sensing problem for small dimensions. One of them, additionally based on branch-and-bound, when applied to image registration under affine transformations, performs on par with or outperforms state-of-the-art methods on benchmark datasets.
Let $I_1,dots,I_n$ be ideals generated by linear forms in a polynomial ring over an infinite field and let $J = I_1 cdots I_n$. We describe a minimal free resolution of $J$ and show that it is supported on a polymatroid obtained from the underlying r
epresentable polymatroid by means of the so-called Dilworth truncation. Formulas for the projective dimension and Betti numbers are given in terms of the polymatroid as well as a characterization of the associated primes. Along the way we show that $J$ has linear quotients. In fact, we do this for a large class of ideals $J_P$, where $P$ is a certain poset ideal associated to the underlying subspace arrangement.
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We introduce a new algebra associated with a hyperplane arrangement $mathcal{A}$, called the Solomon-Terao algebra $mbox{ST}(mathcal{A},eta)$, where $eta$ is a homogeneous polynomial. It is shown by Solomon and Terao that $mbox{ST}(mathcal{A},eta)$ i
s Artinian when $eta$ is generic. This algebra can be considered as a generalization of coinvariant algebras in the setting of hyperplane arrangements. The class of Solomon-Terao algebras contains cohomology rings of regular nilpotent Hessenberg varieties. We show that $mbox{ST}(mathcal{A},eta)$ is a complete intersection if and only if $mathcal{A}$ is free. We also give a factorization formula of the Hilbert polynomials when $mathcal{A}$ is free, and pose several related questions, problems and conjectures.
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