21,452 research outputs found
The Euclidean distance degree of an algebraic variety
The nearest point map of a real algebraic variety with respect to Euclidean
distance is an algebraic function. For instance, for varieties of low rank
matrices, the Eckart-Young Theorem states that this map is given by the
singular value decomposition. This article develops a theory of such nearest
point maps from the perspective of computational algebraic geometry. The
Euclidean distance degree of a variety is the number of critical points of the
squared distance to a generic point outside the variety. Focusing on varieties
seen in applications, we present numerous tools for exact computations.Comment: to appear in Foundations of Computational Mathematic
Polar degrees and closest points in codimension two
Suppose that is a toric variety of codimension
two defined by an integer matrix , and let be a Gale
dual of . In this paper we compute the Euclidean distance degree and polar
degrees of (along with other associated invariants) combinatorially
working from the matrix . Our approach allows for the consideration of
examples that would be impractical using algebraic or geometric methods. It
also yields considerably simpler computational formulas for these invariants,
allowing much larger examples to be computed much more quickly than the
analogous combinatorial methods using the matrix in the codimension two
case.Comment: 25 pages, 1 figur
Asymptotics of degrees and ED degrees of Segre products
Two fundamental invariants attached to a projective variety are its classical algebraic degree and its Euclidean Distance degree (ED degree). In this paper, we study the asymptotic behavior of these two degrees of some Segre products and their dual varieties. We analyze the asymptotics of degrees of (hypercubical) hyperdeterminants, the dual hypersurfaces to Segre varieties. We offer an alternative viewpoint on the stabilization of the ED degree of some Segre varieties. Although this phenomenon was incidentally known from Friedland-Ottaviani's formula expressing the number of singular vector tuples of a general tensor, our approach provides a geometric explanation. Finally, we establish the stabilization of the degree of the dual variety of a Segre product X×Qn, where X is a projective variety and Qn⊂Pn+1 is a smooth quadric hypersurface
The algebraic degree of sparse polynomial optimization
In this paper we study a broad class of polynomial optimization problems
whose constraints and objective functions exhibit sparsity patterns. We give
two characterizations of the number of critical points to these problems, one
as a mixed volume and one as an intersection product on a toric variety. As a
corollary, we obtain a convex geometric interpretation of polar degrees, a
classical invariant of algebraic varieties as well as Euclidean distance
degrees. Furthermore, we prove BKK generality of Lagrange systems in many
instances. Motivated by our result expressing the algebraic degree of sparse
polynomial optimisation problems via Porteus' formula, in the appendix we
answer a related question concerning the degree of sparse determinantal
varieties.Comment: 29 page
Algebra and Geometry of Camera Resectioning
We study algebraic varieties associated with the camera resectioning problem.
We characterize these resectioning varieties' multigraded vanishing ideals
using Gr\"obner basis techniques. As an application, we derive and re-interpret
celebrated results in geometric computer vision related to camera-point
duality. We also clarify some relationships between the classical problems of
optimal resectioning and triangulation, state a conjectural formula for the
Euclidean distance degree of the resectioning variety, and discuss how this
conjecture relates to the recently-resolved multiview conjecture.Comment: 27 page
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