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Paul Breiding

Publications and source records attributed to Paul Breiding.

At least 37 records · Page 2Linked to original sources
arXiv2022

Line Multiview Varieties

We present an algebraic study of line correspondences for pinhole cameras, in contrast to the thoroughly studied point correspondences. We define the line multiview variety as the Zariski closure of the image of the map projecting lines in 3-space to tuples of image lines in 2-space. We prove that in the case of generic camera matrices the line multiview variety is a determinantal variety and we provide a complete set-theoretic description for any camera arrangement. We investigate basic properties of this variety such as dimension, smoothness, and multidegree. Finally, we give experimental results for the Euclidean distance degree and robustness under noise for the triangulation of lines.

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arXiv2022

Real circles tangent to 3 conics

In this paper we study circles tangent to conics. We show there are generically $184$ complex circles tangent to three conics in the plane and we characterize the real discriminant of the corresponding polynomial system. We give an explicit example of $3$ conics with $136$ real circles tangent to them. We conjecture that 136 is the maximal number of real circles. Furthermore, we implement a hill-climbing algorithm to find instances of conics with many real circles, and we introduce a machine learning model that, given three real conics, predicts the number of circles tangent to these three conics.

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arXiv2022

The average condition number of most tensor rank decomposition problems is infinite

The tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with respect to structured perturbations. Those are perturbations preserving the rank of the tensor that is decomposed. On the other hand, the angular condition number measures the perturbations of the rank-1 summands up to scaling. We show for random rank-2 tensors that the expected value of the condition number is infinite for a wide range of choices of the density. Under a mild additional assumption, we show that the same is true for most higher ranks $r\geq 3$ as well. In fact, as the dimensions of the tensor tend to infinity, asymptotically all ranks are covered by our analysis. On the contrary, we show that rank-2 tensors have finite expected angular condition number. Based on numerical experiments, we conjecture that this could also be true for higher ranks. Our results underline the high computational complexity of computing tensor rank decompositions. We discuss consequences of our results for algorithm design and for testing algorithms computing tensor rank decompositions.

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arXiv2022

The Algebraic Degree of Coupled Oscillators

Approximating periodic solutions to the coupled Duffing equations amounts to solving a system of polynomial equations. The number of complex solutions measures the algebraic complexity of this approximation problem. Using the theory of Khovanskii bases, we show that this number is given by the volume of a certain polytope. We also show how to compute all solutions using numerical nonlinear algebra.

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arXiv2022

The zonoid algebra, generalized mixed volumes, and random determinants

We show that every multilinear map between Euclidean spaces induces a unique, continuous, Minkowski multilinear map of the corresponding real cones of zonoids. Applied to the wedge product of the exterior algebra of a Euclidean space, this yields a multiplication of zonoids, defining the structure of a commutative, associative, and partially ordered ring, which we call the zonoid algebra. This framework gives a new perspective on classical objects in convex geometry, and it allows to introduce new functionals on zonoids, in particular generalizing the notion of mixed volume. We also analyze a similar construction based on the complex wedge product, which leads to the new notion of mixed $J$-volume. These ideas connect to the theory of random determinants.

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arXiv2021

Facet volumes of polytopes

In this paper, motivated by the work of Edelman and Strang, we show that for fixed integers $d\geq 2$ and $n\geq d+1$ the configuration space of all facet volume vectors of all $d$-polytopes in $\mathbb R^{d}$ with $n$ facets is a full dimensional cone in $\mathbb R^{n}$. In particular, for tetrahedra ($d=3$ and $n=4$) this is a cone over a regular octahedron. Our proof is based on a novel configuration space / test map scheme which uses topological methods for finding solutions of a problem, and tools of differential geometry to identify solutions with the desired properties. Furthermore, our results open a possibility for the study of realization spaces of all $d$-polytopes in $\mathbb R^{d}$ with $n$ facets by the methods of algebraic topology.

math.CO↗
arXiv2021

Equations for GL invariant families of polynomials

We provide an algorithm that takes as an input a given parametric family of homogeneous polynomials, which is invariant under the action of the general linear group, and an integer $d$. It outputs the ideal of that family intersected with the space of homogeneous polynomials of degree $d$. Our motivation comes from open problems, which ask to find equations for varieties of cubic and quartic symmetroids. The algorithm relies on a database of specific Young tableaux and highest weight polynomials. We provide the database and the implementation of the database construction algorithm. Moreover, we provide a julia implementation to run the algorithm using the database, so that more varieties of homogeneous polynomials can easily be treated in the future.

math.AC↗
arXiv2021

Three decompositions of symmetric tensors have similar condition numbers

We relate the condition numbers of computing three decompositions of symmetric tensors: the canonical polyadic decomposition, the Waring decomposition, and a Tucker-compressed Waring decomposition. Based on this relation we can speed up the computation of these condition numbers by orders of magnitude

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arXiv2021

Nonlinear Algebra and Applications

We showcase applications of nonlinear algebra in the sciences and engineering. Our review is organized into eight themes: polynomial optimization, partial differential equations, algebraic statistics, integrable systems, configuration spaces of frameworks, biochemical reaction networks, algebraic vision, and tensor decompositions. Conversely, developments on these topics inspire new questions and algorithms for algebraic geometry.

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arXiv2021

Sensitivity of low-rank matrix recovery

We characterize the first-order sensitivity of approximately recovering a low-rank matrix from linear measurements, a standard problem in compressed sensing. A special case covered by our analysis is approximating an incomplete matrix by a low-rank matrix. We give an algorithm for computing the associated condition number and demonstrate experimentally how the number of linear measurements affects it. In addition, we study the condition number of the rank-r matrix approximation problem. It measures in the Frobenius norm by how much an infinitesimal perturbation to an arbitrary input matrix is amplified in the movement of its best rank-r approximation. We give an explicit formula for the condition number, which shows that it does depend on the relative singular value gap between the rth and (r+1)th singular values of the input matrix.

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arXiv2021

The condition number of many tensor decompositions is invariant under Tucker compression

We characterise the sensitivity of several additive tensor decompositions with respect to perturbations of the original tensor. These decompositions include canonical polyadic decompositions, block term decompositions, and sums of tree tensor networks. Our main result shows that the condition number of all these decompositions is invariant under Tucker compression. This result can dramatically speed up the computation of the condition number in practical applications. We give the example of an $265\times 371\times 7$ tensor of rank $3$ from a food science application whose condition number was computed in $6.9$ milliseconds by exploiting our new theorem, representing a speedup of four orders of magnitude over the previous state of the art.

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arXiv2021

Euclidean distance degree and mixed volume

We initiate a study of the Euclidean Distance Degree in the context of sparse polynomials. Specifically, we consider a hypersurface f=0 defined by a polynomial f that is general given its support, such that the support contains the origin. We show that the Euclidean Distance Degree of f=0 equals the mixed volume of the Newton polytopes of the associated Lagrange multiplier equations. We discuss the implication of our result for computational complexity and give a formula for the Euclidean distance degree when the Newton polytope is a rectangular parallelepiped.

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arXiv2020

The condition number of Riemannian approximation problems

We consider the local sensitivity of least-squares formulations of inverse problems. The sets of inputs and outputs of these problems are assumed to have the structures of Riemannian manifolds. The problems we consider include the approximation problem of finding the nearest point on a Riemannian embedded submanifold from a given point in the ambient space. We characterize the first-order sensitivity, i.e., condition number, of local minimizers and critical points to arbitrary perturbations of the input of the least-squares problem. This condition number involves the Weingarten map of the input manifold, which measures the amount by which the input manifold curves in its ambient space. We validate our main results through experiments with the $n$-camera triangulation problem in computer vision.

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arXiv2020

Quantitative singularity theory for random polynomials

Motivated by Hilbert's 16th problem we discuss the probabilities of topological features of a system of random homogeneous polynomials. The distribution for the polynomials is the Kostlan distribution. The topological features we consider are type-$W$ singular loci. This is a term that we introduce and that is defined by a list of equalities and inequalities on the derivatives of the polynomials. In technical terms a type-$W$ singular locus is the set of points where the jet of the Kostlan polynomials belongs to a semialgebraic subset $W$ of the jet space, which we require to be invariant under orthogonal change of variables. For instance, the zero set of polynomial functions or the set of critical points fall under this definition. We will show that, with overwhelming probability, the type-$W$ singular locus of a Kostlan polynomial is ambient isotopic to that of a polynomial of lower degree. As a crucial result, this implies that complicated topological configurations are rare. Our results extend earlier results from Diatta and Lerario who considered the special case of the zero set of a single polynomial. Furthermore, for a given polynomial function $p$ we provide a deterministic bound for the radius of the ball in the space of differentiable functions with center $p$, in which the $W$-singularity structure is constant.

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arXiv2020

Random points on an algebraic manifold

Consider the set of solutions to a system of polynomial equations in many variables. An algebraic manifold is an open submanifold of such a set. We introduce a new method for computing integrals and sampling from distributions on algebraic manifolds. This method is based on intersecting with random linear spaces. It produces i.i.d. samples, works in the presence of multiple connected components, and is simple to implement. We present applications to computational statistical physics and topological data analysis.

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arXiv2019

Random spectrahedra

Spectrahedra are affine-linear sections of the cone $\mathcal{P}_n$ of positive semidefinite symmetric $n\times n$-matrices. We consider random spectrahedra that are obtained by intersecting~$\mathcal{P}_n$ with the affine-linear space $\mathbf{1} + V$, where $\mathbf{1}$ is the identity matrix and $V$ is an $\ell$-dimensional linear space that is chosen from the unique orthogonally invariant probability measure on the Grassmanian of $\ell$-planes in the space of $n\times n$ real symmetric matrices (endowed with the Frobenius inner product). Motivated by applications, for $\ell=3$ we relate the average number $\mathbb{E} σ_n$ of singular points on the boundary of a three-dimensional spectrahedron to the volume of the set of symmetric matrices whose two smallest eigenvalues coincide. In the case of quartic spectrahedra ($n=4$) we show that $\mathbb{E} σ_4 = 6-\frac{4}{\sqrt{3}}$. Moreover, we prove that the average number $\mathbb{E} ρ_n$ of singular points on the real variety of singular matrices in $\mathbf{1} + V$ is $n(n-1)$. This quantity is related to the volume of the variety of real symmetric matrices with repeated eigenvalues. Furthermore, we compute the asymptotics of the volume and the volume of the boundary of a random spectrahedron.

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arXiv2019

3264 Conics in a Second

Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical algebraic geometry determines these solutions for any given instance. This article illustrates how these two fields complement each other. Our focus lies on the 3264 conics that are tangent to five given conics in the plane. We present a web interface for computing them. It uses the software HomotopyContinuation.jl, which makes this process fast and reliable. We discuss an instance where all 3264 solutions are real.

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