Abstract. A complex frame is a collection of vectors that span C M and define measurements, called intensity measurements, on vectors in C M . In purely mathematical terms, the problem of phase retrieval is to recover a complex vector from its intensity measurements, namely the modulus of its inner product with these frame vectors. We show that any vector is uniquely determined (up to a global phase factor) from 4M − 4 generic measurements. To prove this, we identify the set of frames defining non-injective measurements with the projection of a real variety and bound its dimension.
We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where 0 < q < 1. Consequently, we can sample random spanning forests in a graph and (approximately) compute the reliability polynomial of any matroid. We also prove the thirty year old conjecture of Mihail and Vazirani that the bases exchange graph of any matroid has expansion at least 1.Our algorithm and the proof build on the recent results of Dinur, Kaufman, Mass and Oppenheim [KM17; DK17; KO18] who show that high dimensional walks on simplicial complexes mix rapidly if the corresponding localized random walks on 1-skeleton of links of all complexes are strong spectral expanders. One of our key observations is a close connection between pure simplicial complexes and multiaffine homogeneous polynomials. Specifically, if X is a pure simplicial complex with positive weights on its maximal faces, we can associate with X a multiaffine homogeneous polynomial p X such that the eigenvalues of the localized random walks on X correspond to the eigenvalues of the Hessian of derivatives of p X .
A smooth quartic curve in the complex projective plane has 36 inequivalent representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. These correspond to Cayley octads and Steiner complexes respectively. We present exact algorithms for computing these objects from the 28 bitangents. This expresses Vinnikov quartics as spectrahedra and positive quartics as Gram matrices. We explore the geometry of Gram spectrahedra and we find equations for the variety of Cayley octads. Interwoven is an exposition of much of the 19th century theory of plane quartics.
We give a deterministic polynomial time 2 O(r) -approximation algorithm for the number of bases of a given matroid of rank r and the number of common bases of any two matroids of rank r. To the best of our knowledge, this is the first nontrivial deterministic approximation algorithm that works for arbitrary matroids. Based on a lower bound of Azar, Broder, and Frieze [ABF94] this is almost the best possible result assuming oracle access to independent sets of the matroid.There are two main ingredients in our result: For the first, we build upon recent results of Adiprasito, Huh, and Katz [AHK15] and Huh and Wang [HW17] on combinatorial hodge theory to derive a connection between matroids and log-concave polynomials. We expect that several new applications in approximation algorithms will be derived from this connection in future. Formally, we prove that the multivariate generating polynomial of the bases of any matroid is log-concave as a function over the positive orthant. For the second ingredient, we develop a general framework for approximate counting in discrete problems, based on convex optimization. The connection goes through subadditivity of the entropy. For matroids, we prove that an approximate superadditivity of the entropy holds by relying on the logconcavity of the corresponding polynomials.
If a real symmetric matrix of linear forms is positive definite at some point, then its determinant defines a hyperbolic hypersurface. In 2007, Helton and Vinnikov proved a converse in three variables, namely that every hyperbolic curve in the projective plane has a definite real symmetric determinantal representation. The goal of this paper is to give a more concrete proof of a slightly weaker statement. Here we show that every hyperbolic plane curve has a definite determinantal representation with Hermitian matrices. We do this by relating the definiteness of a matrix to the real topology of its minors and extending a construction of Dixon from 1902. Like the Helton-Vinnikov theorem, this implies that every hyperbolic region in the plane is defined by a linear matrix inequality. arXiv:1207.7047v2 [math.AG]
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