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Rainer Sinn

Publications and source records attributed to Rainer Sinn.

At least 37 records · Page 2Linked to original sources
arXiv2021

Existence of Two View Chiral Reconstructions

A fundamental question in computer vision is whether a set of point pairs is the image of a scene that lies in front of two cameras. Such a scene and the cameras together are known as a chiral reconstruction of the point pairs. In this paper we provide a complete classification of k point pairs for which a chiral reconstruction exists. The existence of chiral reconstructions is equivalent to the non-emptiness of certain semialgebraic sets. For up to three point pairs, we prove that a chiral reconstruction always exists while the set of five or more point pairs that do not have a chiral reconstruction is Zariski-dense. We show that for five generic point pairs, the chiral region is bounded by line segments in a Schläfli double six on a cubic surface with 27 real lines. Four point pairs have a chiral reconstruction unless they belong to two non-generic combinatorial types, in which case they may or may not.

cs.CV↗
arXiv2021

Maximum Likelihood Estimation for Nets of Conics

We study the problem of maximum likelihood estimation for $3$-dimensional linear spaces of $3\times 3$ symmetric matrices from the point of view of algebraic statistics where we view these nets of conics as linear concentration or linear covariance models of Gaussian distributions on $\mathbb{R}^3$. In particular, we study the reciprocal surfaces of nets of conics which are rational surfaces in $\mathbb{P}^5$. We show that the reciprocal surfaces are projections from the Veronese surface and determine their intersection with the polar nets. This geometry explains the maximum likelihood degrees of these linear models. We compute the reciprocal maximum likelihood degrees. This work is based on Wall's classification of nets of conics from 1977.

math.AG↗
arXiv2021

Do alcoved lattice polytopes have unimodal h*-vector?

We show that h*-vectors of alcoved polytopes P in R^n (of Lie type A) are unimodal if they contain interior lattice points and their facets have lattice distance 1 to the set of interior lattice points. The maximal possible such distance for general alcoved polytopes is shown to be dim(P)-1. A secondary purpose of the paper is to serve as a guide to previous work surrounding unimodality of h*-vectors of alcoved polytopes and related questions.

math.CO↗
arXiv2021

Families of faces and the normal cycle of a convex semi-algebraic set

We study families of faces for convex semi-algebraic sets via the normal cycle which is a semi-algebraic set similar to the conormal variety in projective duality theory. We propose a convex algebraic notion of a "patch" -- a term recently coined by Ciripoi, Kaihnsa, Löhne, and Sturmfels as a tool for approximating the convex hull of a semi-algebraic set. We discuss geometric consequences, both for the semi-algebraic and convex geometry of the families of faces, as well as variations of our definition and their consequences.

math.AG↗
arXiv2021

Sums of Squares: A Real Projective Story

The relationship between nonnegative polynomials and sums of squares is one of the central questions in real algebraic geometry. A modern approach is to look at nonnegative polynomials and sums of squares on a real variety. We survey the rich history of this area and focus on recent developments highlighting unexpected links to complex algebraic geometry and commutative algebra.

math.AG↗
arXiv2020

On the Dimensions of the Realization Spaces of Polytopes

Robertson (1988) suggested a model for the realization space of a convex d-dimensional polytope and an approach via the implicit function theorem, which -- in the case of a full rank Jacobian -- proves that the realization space is a manifold of dimension NG(P):=d(f_0+f_{d-1})-f_{0,d-1}, which is the natural guess for the dimension given by the number of variables minus the number of quadratic equations that are used in the definition of the realization space. While this indeed holds for many natural classes of polytopes (including simple and simplicial polytopes, as well as all polytopes of dimension at most 3),and Robertson claimed this to be true for all polytopes, Mnev's (1986/1988) Universality Theorem implies that it is not true in general: Indeed, (1) the centered realization space is not a smoothly embedded manifold in general, and (2) it does not have the dimension NG(P) in general. In this paper we develop Jacobian criteria for the analysis of realization spaces. From these we get easily that for various large and natural classes of polytopes the realization spaces are indeed manifolds, whose dimensions are given by NG(P). However, we also identify the smallest polytopes where the dimension count (2) and thus Robertson's claim fails, among them the bipyramid over a triangular prism. For the property (1), we analyze the classical 24-cell: We show that the realization space has at least the dimension 48, and it has points where it is a manifold of this dimension, but it is not smoothly embedded as a manifold everywhere.

math.MG↗
arXiv2020

Kippenhahn's Theorem for joint numerical ranges and quantum states

Kippenhahn's Theorem asserts that the numerical range of a matrix is the convex hull of a certain algebraic curve. Here, we show that the joint numerical range of finitely many Hermitian matrices is similarly the convex hull of a semi-algebraic set. We discuss an analogous statement regarding the dual convex cone to a hyperbolicity cone and prove that the class of bases of these dual cones is closed under linear operations. The result offers a new geometric method to analyze quantum states.

math.AG↗
arXiv2020

Sums of Squares and Quadratic Persistence on Real Projective Varieties

We bound the Pythagoras number of a real projective subvariety: the smallest positive integer $r$ such that every sum of squares of linear forms in its homogeneous coordinate ring is a sum of at most $r$ squares. Enhancing existing methods, we exhibit three distinct upper bounds involving known invariants. In contrast, our lower bound depends on a new invariant of a projective subvariety called the quadratic persistence. Defined by projecting away from points, this numerical invariant is closely related to the linear syzygies of the variety. In addition, we classify the projective subvarieties of maximal and almost-maximal quadratic persistence, and determine their Pythagoras numbers.

math.AG↗
arXiv2020

Hyperbolic Secant Varieties of M-Curves

We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibered morphism to a projective space whose dimension is equal to the dimension of the variety. We study hyperbolic varieties with a special interest in the case of hypersurfaces that admit a real algebraic ruling. The central part of the paper is concerned with secant varieties of real algebraic curves where the real locus has the maximal number of connected components, which is determined by the genus of the curve. For elliptic normal curves, we further obtain definite symmetric determinantal representations for the hyperbolic secant hypersurfaces, which implies the existence of symmetric Ulrich sheaves of rank one on these hypersurfaces.

math.AG↗
arXiv2019

Combinatorial Inscribability Obstructions for Higher-Dimensional Polytopes

For $3$-dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hyperbolic geometry. In particular, inscribability can be tested in polynomial time, and for every $f$-vector of $3$-polytopes, there exists an inscribable polytope with that $f$-vector. For higher-dimensional polytopes, much less is known. Of course, for any inscribable polytope, all of its lower-dimensional faces need to be inscribable, but this condition does not appear to be very strong. We observe non-trivial new obstructions to the inscribability of polytopes that arise when imposing that a certain inscribable face be inscribed. Using this obstruction, we show that the duals of the $4$-dimensional cyclic polytopes with at least $8$ vertices---all of whose faces are inscribable---are not inscribable. This result is optimal in the following sense: We prove that the duals of the cyclic $4$-polytopes with up to $7$ vertices are, in fact, inscribable. Moreover, we interpret this obstruction combinatorially as a forbidden subposet of the face lattice of a polytope, show that $d$-dimensional cyclic polytopes with at least $d+4$ vertices are not circumscribable, and that no dual of a neighborly $4$-polytope with $8$ vertices, that is, no polytope with $f$-vector $(20,40,28,8)$, is inscribable.

math.MG↗
arXiv2019

Typical and Generic Ranks in Matrix Completion

We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrix M, we keep the positions of specified and unspecified entries fixed, and study how the minimal completion rank depends on the values of the known entries. If the entries of the matrix are complex numbers, then for a fixed pattern of locations of specified and unspecified entries there is a unique completion rank which occurs with positive probability. We call this rank the generic completion rank. Over the real numbers there can be multiple ranks that occur with positive probability; we call them typical completion ranks. We introduce these notions formally, and provide a number of inequalities and exact results on typical and generic ranks for different families of patterns of known and unknown entries.

math.ST↗
arXiv2019

Conic programming: infeasibility certificates and projective geometry

We revisit facial reduction from the point of view of projective geometry. This leads us to a homogenization strategy in conic programming that eliminates the phenomenon of weak infeasibility. For semidefinite programs (and others), this yields infeasibility certificates that can be checked in polynomial time. Furthermore, we propose a refined type of infeasibility, which we call stably infeasible, for which rational infeasibility certificates exist and that can be distinguished from other infeasibility types by our homogenization.

math.OC↗
arXiv2018

Gram Spectrahedra

Representations of nonnegative polynomials as sums of squares are central to real algebraic geometry and the subject of active research. The sum-of-squares representations of a given polynomial are parametrized by the convex body of positive semidefinite Gram matrices, called the Gram spectrahedron. This is a fundamental object in polynomial optimization and convex algebraic geometry. We summarize results on sums of squares that fit naturally into the context of Gram spectrahedra, present some new results, and highlight related open questions. We discuss sum-of-squares representations of minimal length and relate them to Hermitian Gram spectrahedra and point evaluations on toric varieties.

math.AG↗
arXiv2017

Positive Semidefinite Univariate Matrix Polynomials

We study sum-of-squares representations of symmetric univariate real matrix polynomials that are positive semidefinite along the real line. We give a new proof of the fact that every positive semidefinite univariate matrix polynomial of size $n\times n$ can be written as a sum of squares $M=Q^TQ$, where $Q$ has size $(n+1)\times n$, which was recently proved by Blekherman-Plaumann-Sinn-Vinzant. Our new approach using the theory of quadratic forms allows us to prove the conjecture made by these authors that these minimal representations $M=Q^TQ$ are generically in one-to-one correspondence with the representations of the nonnegative univariate polynomial $\det(M)$ as sums of two squares. In parallel, we will use our methods to prove the more elementary hermitian analogue that every hermitian univariate matrix polynomial $M$ that is positive semidefinite along the real line, is a square, which is known as the matrix Fejér-Riesz Theorem.

math.AG↗
arXiv2017

Maximum Likelihood Threshold and Generic Completion Rank of Graphs

The minimum number of observations such that the maximum likelihood estimator in a Gaussian graphical model exists with probability one is called the maximum likelihood threshold of the underlying graph G. The natural algebraic relaxation is the generic completion rank introduced by Uhler. We show that the maximum likelihood threshold and the generic completion rank behave in the same way under clique sums, which gives us large families of graphs on which these invariants coincide. On the other hand we determine both invariants for complete bipartite graphs $K_{m,n}$ and show that for some choices of m and n the two parameters may be quite far apart. In particular, this gives the first examples of graphs on which the maximum likelihood threshold and the generic completion rank do not agree.

math.CO↗
arXiv2017

Low-Rank Sum-of-Squares Representations on Varieties of Minimal Degree

A celebrated result by Hilbert says that every real nonnegative ternary quartic is a sum of three squares. We show more generally that every nonnegative quadratic form on a real projective variety $X$ of minimal degree is a sum of $\dim(X)+1$ squares of linear forms. This strengthens one direction of a recent result due to Blekherman, Smith, and Velasco. Our upper bound is the best possible, and it implies the existence of low-rank factorizations of positive semidefinite bivariate matrix polynomials and representations of biforms as sums of few squares. We determine the number of equivalence classes of sum-of-squares representations of general quadratic forms on surfaces of minimal degree, generalizing the count for ternary quartics by Powers, Reznick, Scheiderer, and Sottile.

math.AG↗
arXiv2016

Do Sums of Squares Dream of Free Resolutions?

We associate to a real projective variety $X$ two convex cones which are fundamental in real algebraic geometry: the cone $P_X$ of quadratic forms nonnegative on $X$, and the cone $Σ_X$ of sums of squares of linear forms. The dual cone $Σ_X^\ast$ is a spectrahedron and we show that its convexity properties are closely related to homological properties of $X$. For instance, we show that all extreme rays of $Σ_X^\ast$ have rank one if and only if X has Castelnuovo-Mumford regularity two. More generally, if $Σ_X^\ast$ has an extreme ray of rank $p > 1$, then $X$ does not satisfy the property $N_{2,p}$. We show that the converse also holds in a wide variety of situations: the smallest $p$ for which property $N_{2,p}$ does not hold is equal to the smallest rank of an extreme ray of $Σ_X^\ast$ greater than one. These results allow us to generalize the work of Blekherman-Smith-Velasco on equality of nonnegative polynomials and sums of squares from irreducible varieties to reduced schemes and to classify all spectrahedral cones with only rank one extreme rays. Our results have applications to the positive semidefinite matrix completion problem and to the truncated moment problem on projective varieties.

math.AG↗