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Register a new userBad Projections of the PSD Cone
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Added by
Yuhan Jiang
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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The image of the cone of positive semidefinite matrices under a linear map is a convex cone. Pataki characterized the set of linear maps for which that image is not closed. The Zariski closure of this set is a hypersurface in the Grassmannian. Its components are the coisotropic hypersurfaces of symmetric determinantal varieties. We develop the convex algebraic geometry of such bad projections, with focus on explicit computations.
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We study the problem of approximating the cone of positive semidefinite (PSD) matrices with a cone that can be described by smaller-sized PSD constraints. Specifically, we ask the question: how closely can we approximate the set of unit-trace $n times n$ PSD matrices, denoted by $D$, using at most $N$ number of $k times k$ PSD constraints? In this paper, we prove lower bounds on $N$ to achieve a good approximation of $D$ by considering two constructions of an approximating set. First, we consider the unit-trace $n times n$ symmetric matrices that are PSD when restricted to a fixed set of $k$-dimensional subspaces in $mathbb{RR}^n$. We prove that if this set is a good approximation of $D$, then the number of subspaces must be at least exponentially large in $n$ for any $k = o(n)$. % Second, we show that any set $S$ that approximates $D$ within a constant approximation ratio must have superpolynomial $mathbf{S}_+^k$-extension complexity. To be more precise, if $S$ is a constant factor approximation of $D$, then $S$ must have $mathbf{S}_+^k$-extension complexity at least $exp( C cdot min { sqrt{n}, n/k })$ where $C$ is some absolute constant. In addition, we show that any set $S$ such that $D subseteq S$ and the Gaussian width of $D$ is at most a constant times larger than the Gaussian width of $D$ must have $mathbf{S}_+^k$-extension complexity at least $exp( C cdot min { n^{1/3}, sqrt{n/k} })$. These results imply that the cone of $n times n$ PSD matrices cannot be approximated by a polynomial number of $k times k$ PSD constraints for any $k = o(n / log^2 n)$. These results generalize the recent work of Fawzi on the hardness of polyhedral approximations of $mathbf{S}_+^n$, which corresponds to the special case with $k=1$.
A bad point of a positive semidefinite real polynomial f is a point at which a pole appears in all expressions of f as a sum of squares of rational functions. We show that quartic polynomials in three variables never have bad points. We give examples of positive semidefinite polynomials with a bad point at the origin, that are nevertheless sums of squares of formal power series, answering a question of Brumfiel. We also give an example of a positive semidefinite polynomial in three variables with a complex bad point that is not real, answering a question of Scheiderer.
We study the ramification divisors of projections of a smooth projective variety onto a linear subspace of the same dimension. We prove that the ramification divisors vary in a maximal dimensional family for a large class of varieties. Going further, we study the map that associates to a linear projection its ramification divisor. We show that this map is dominant for most (but not all!) varieties of minimal degree, using (linked) limit linear series of higher rank. We find the degree of this map in some cases, extending the classical appearance of Catalan numbers in the geometry of rational normal curves, and give a geometric explanation of its fibers in terms of torsion points of naturally occurring elliptic curves in the case of the Veronese surface and the quartic rational surface scroll.
Let X be a smooth, connected, closed subvariety of a complex vector space V. The asymptotic cone as(X) is naturally equipped with a nearby cycles sheaf P coming from the specialization of X to as(X). We show that if X is transverse to infinity in a suitable sense, then the Fourier transform of P is an intersection homology sheaf.
Patient respiratory signal associated with the cone beam CT (CBCT) projections is important for lung cancer radiotherapy. In contrast to monitoring an external surrogate of respiration, such signal can be extracted directly from the CBCT projections. In this paper, we propose a novel local principle component analysis (LPCA) method to extract the respiratory signal by distinguishing the respiration motion-induced content change from the gantry rotation-induced content change in the CBCT projections. The LPCA method is evaluated by comparing with three state-of-the-art projection-based methods, namely, the Amsterdam Shroud (AS) method, the intensity analysis (IA) method, and the Fourier-transform based phase analysis (FT-p) method. The clinical CBCT projection data of eight patients, acquired under various clinical scenarios, were used to investigate the performance of each method. We found that the proposed LPCA method has demonstrated the best overall performance for cases tested and thus is a promising technique for extracting respiratory signal. We also identified the applicability of each existing method.
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