arXiv subjects

Elias Tsigaridas

Publications and source records attributed to Elias Tsigaridas.

At least 37 records · Page 2Linked to original sources

arXiv2022

Trifocal Relative Pose from Lines at Points and its Efficient Solution

We present a method for solving two minimal problems for relative camera pose estimation from three views, which are based on three view correspondences of i) three points and one line and the novel case of ii) three points and two lines through two of the points. These problems are too difficult to be efficiently solved by the state of the art Groebner basis methods. Our method is based on a new efficient homotopy continuation (HC) solver framework MINUS, which dramatically speeds up previous HC solving by specializing HC methods to generic cases of our problems. We characterize their number of solutions and show with simulated experiments that our solvers are numerically robust and stable under image noise, a key contribution given the borderline intractable degree of nonlinearity of trinocular constraints. We show in real experiments that i) SIFT feature location and orientation provide good enough point-and-line correspondences for three-view reconstruction and ii) that we can solve difficult cases with too few or too noisy tentative matches, where the state of the art structure from motion initialization fails.

cs.CV↗

arXiv2022

One-connection rule for structural equation models

Linear structural equation models are multivariate statistical models encoded by mixed graphs. In particular, the set of covariance matrices for distributions belonging to a linear structural equation model for a fixed mixed graph $G=(V, D,B)$ is parameterized by a rational function with parameters for each vertex and edge in $G$. This rational parametrization naturally allows for the study of these models from an algebraic and combinatorial point of view. Indeed, this point of view has led to a collection of results in the literature, mainly focusing on questions related to identifiability and determining relationships between covariances (i.e., finding polynomials in the Gaussian vanishing ideal). So far, a large proportion of these results has focused on the case when $D$, the directed part of the mixed graph $G$, is acyclic. This is due to the fact that in the acyclic case, the parametrization becomes polynomial and there is a description of the entries of the covariance matrices in terms of a finite sum. We move beyond the acyclic case and give a closed form expression for the entries of the covariance matrices in terms of the one-connections in a graph obtained from $D$ through some small operations. This closed form expression then allows us to show that if $G$ is simple, then the parametrization map is generically finite-to-one. Finally, having a closed form expression for the covariance matrices allows for the development of an algorithm for systematically exploring possible polynomials in the Gaussian vanishing ideal.

math.ST↗

arXiv2022

Beyond Worst-Case Analysis for Root Isolation Algorithms

Isolating the real roots of univariate polynomials is a fundamental problem in symbolic computation and it is arguably one of the most important problems in computational mathematics. The problem has a long history decorated with numerous ingenious algorithms and furnishes an active area of research. However, the worst-case analysis of root-finding algorithms does not correlate with their practical performance. We develop a smoothed analysis framework for polynomials with integer coefficients to bridge the gap between the complexity estimates and the practical performance. In this setting, we derive that the expected bit complexity of DESCARTES solver to isolate the real roots of a polynomial, with coefficients uniformly distributed, is $\widetilde{\mathcal{O}}_B(d^2 + d τ)$, where $d$ is the degree of the polynomial and $τ$ the bitsize of the coefficients.

cs.CC↗

arXiv2022

Randomized geometric tools for anomaly detection in stock markets

We propose novel randomized geometric tools to detect low-volatility anomalies in stock markets; a principal problem in financial economics. Our modeling of the (detection) problem results in sampling and estimating the (relative) volume of geodesically non-convex and non-connected spherical patches that arise by intersecting a non-standard simplex with a sphere. To sample, we introduce two novel Markov Chain Monte Carlo (MCMC) algorithms that exploit the geometry of the problem and employ state-of-the-art continuous geometric random walks (such as Billiard walk and Hit-and-Run) adapted on spherical patches. To our knowledge, this is the first geometric formulation and MCMC-based analysis of the volatility puzzle in stock markets. We have implemented our algorithms in C++ (along with an R interface) and we illustrate the power of our approach by performing extensive experiments on real data. Our analyses provide accurate detection and new insights into the distribution of portfolios' performance characteristics. Moreover, we use our tools to show that classical methods for low-volatility anomaly detection in finance form bad proxies that could lead to misleading or inaccurate results.

cs.CG↗

arXiv2022

PTOPO: Computing the Geometry and the Topology of Parametric Curves

We consider the problem of computing the topology and describing the geometry of a parametric curve in $\mathbb{R}^n$. We present an algorithm, PTOPO, that constructs an abstract graph that is isotopic to the curve in the embedding space. Our method exploits the benefits of the parametric representation and does not resort to implicitization. Most importantly, we perform all computations in the parameter space and not in the implicit space. When the parametrization involves polynomials of degree at most $d$ and maximum bitsize of coefficients $τ$, then the worst case bit complexity of PTOPO is $ \tilde{\mathcal{O}}_B(nd^6+nd^5τ+d^4(n^2+nτ)+d^3(n^2τ+ n^3)+n^3d^2τ)$. This bound matches the current record bound $\tilde{\mathcal{O}}_B(d^6+d^5τ)$ for the problem of computing the topology of a plane algebraic curve given in implicit form. For plane and space curves, if $N = \max\{d, τ\}$, the complexity of PTOPO becomes $\tilde{\mathcal{O}}_B(N^6)$, which improves the state-of-the-art result, due to Alcázar and Díaz-Toca [CAGD'10], by a factor of $N^{10}$. In the same time complexity, we obtain a graph whose straight-line embedding is isotopic to the curve. However, visualizing the curve on top of the abstract graph construction, increases the bound to $\tilde{\mathcal{O}}_B(N^7)$. For curves of general dimension, we can also distinguish between ordinary and non-ordinary real singularities and determine their multiplicities in the same expected complexity of PTOPO by employing the algorithm of Blasco and Pérez-Díaz [CAGD'19]. We have implemented PTOPO in Maple for the case of plane and space curves. Our experiments illustrate its practical nature.

cs.SC↗

arXiv2022

Exact solutions in low-rank approximation with zeros

Low-rank approximation with zeros aims to find a matrix of fixed rank and with a fixed zero pattern that minimizes the Euclidean distance to a given data matrix. We study the critical points of this optimization problem using algebraic tools. In particular, we describe special linear, affine, and determinantal relations satisfied by the critical points. We also investigate the number of critical points and how this number is related to the complexity of nonnegative matrix factorization problem.

math.AG↗

arXiv2021

GPU-Based Homotopy Continuation for Minimal Problems in Computer Vision

Systems of polynomial equations arise frequently in computer vision, especially in multiview geometry problems. Traditional methods for solving these systems typically aim to eliminate variables to reach a univariate polynomial, e.g., a tenth-order polynomial for 5-point pose estimation, using clever manipulations, or more generally using Grobner basis, resultants, and elimination templates, leading to successful algorithms for multiview geometry and other problems. However, these methods do not work when the problem is complex and when they do, they face efficiency and stability issues. Homotopy Continuation (HC) can solve more complex problems without the stability issues, and with guarantees of a global solution, but they are known to be slow. In this paper we show that HC can be parallelized on a GPU, showing significant speedups up to 26 times on polynomial benchmarks. We also show that GPU-HC can be generically applied to a range of computer vision problems, including 4-view triangulation and trifocal pose estimation with unknown focal length, which cannot be solved with elimination template but they can be efficiently solved with HC. GPU-HC opens the door to easy formulation and solution of a range of computer vision problems.

cs.CV↗

arXiv2021

The Multivariate Schwartz-Zippel Lemma

Motivated by applications in combinatorial geometry, we consider the following question: Let $λ=(λ_1,λ_2,\ldots,λ_m)$ be an $m$-partition of a positive integer $n$, $S_i \subseteq \mathbb{C}^{λ_i}$ be finite sets, and let $S:=S_1 \times S_2 \times \ldots \times S_m \subset \mathbb{C}^n$ be the multi-grid defined by $S_i$. Suppose $p$ is an $n$-variate degree $d$ polynomial. How many zeros does $p$ have on $S$? We first develop a multivariate generalization of Combinatorial Nullstellensatz that certifies existence of a point $t \in S$ so that $p(t) \neq 0$. Then we show that a natural multivariate generalization of the DeMillo-Lipton-Schwartz-Zippel lemma holds, except for a special family of polynomials that we call $λ$-reducible. This yields a simultaneous generalization of Szemerédi-Trotter theorem and Schwartz-Zippel lemma into higher dimensions, and has applications in incidence geometry. Finally, we develop a symbolic algorithm that identifies certain $λ$-reducible polynomials. More precisely, our symbolic algorithm detects polynomials that include a cartesian product of hypersurfaces in their zero set. It is likely that using Chow forms the algorithm can be generalized to handle arbitrary $λ$-reducible polynomials, which we leave as an open problem.

math.CO↗

arXiv2021

Segre-Driven Radicality Testing

We present a probabilistic algorithm to test if a homogeneous polynomial ideal $I$ defining a scheme $X$ in $\mathbb{P}^n$ is radical using Segre classes and other geometric notions from intersection theory. Its worst case complexity depends on the geometry of $X$. If the scheme $X$ has reduced isolated primary components and no embedded components supported the singular locus of $X_{\rm red}=V(\sqrt{I})$, then the worst case complexity is doubly exponential in $n$; in all the other cases the complexity is singly exponential. The realm of the ideals for which our radical testing procedure requires only single exponential time includes examples which are often considered pathological, such as the ones drawn from the famous Mayr-Meyer set of ideals which exhibit doubly exponential complexity for the ideal membership problem.

math.AG↗

arXiv2021

Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces

The condition-based complexity analysis framework is one of the gems of modern numerical algebraic geometry and theoretical computer science. Among the challenges that it poses is to expand the currently limited range of random polynomials that we can handle. Despite important recent progress, the available tools cannot handle random sparse polynomials and Gaussian polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random variables. We initiate a condition-based complexity framework based on the norm of the cube that is a step in this direction. We present this framework for real hypersurfaces and univariate polynomials. We demonstrate its capabilities in two problems, under very mild probabilistic assumptions. On the one hand, we show that the average run-time of the Plantinga-Vegter algorithm is polynomial in the degree for random sparse (alas a restricted sparseness structure) polynomials and random Gaussian polynomials. On the other hand, we study the size of the subdivision tree for Descartes' solver and run-time of the solver by Jindal and Sagraloff (arXiv:1704.06979). In both cases, we provide a bound that is polynomial in the size of the input (size of the support plus the logarithm of the degree) not only for the average but also for all higher moments.

cs.CG↗

arXiv2021

Truncated Log-concave Sampling with Reflective Hamiltonian Monte Carlo

We introduce Reflective Hamiltonian Monte Carlo (ReHMC), an HMC-based algorithm, to sample from a log-concave distribution restricted to a convex body. We prove that, starting from a warm start, the walk mixes to a log-concave target distribution $π(x) \propto e^{-f(x)}$, where $f$ is $L$-smooth and $m$-strongly-convex, within accuracy $\varepsilon$ after $\widetilde O(κd^2 \ell^2 \log (1 / \varepsilon))$ steps for a well-rounded convex body where $κ= L / m$ is the condition number of the negative log-density, $d$ is the dimension, $\ell$ is an upper bound on the number of reflections, and $\varepsilon$ is the accuracy parameter. We also developed an efficient open source implementation of ReHMC and we performed an experimental study on various high-dimensional data-sets. The experiments suggest that ReHMC outperfroms Hit-and-Run and Coordinate-Hit-and-Run regarding the time it needs to produce an independent sample and introduces practical truncated sampling in thousands of dimensions.

cs.LG↗

arXiv2021

Koszul-type determinantal formulas for families of mixed multilinear systems

Effective computation of resultants is a central problem in elimination theory and polynomial system solving. Commonly, we compute the resultant as a quotient of determinants of matrices and we say that there exists a determinantal formula when we can express it as a determinant of a matrix whose elements are the coefficients of the input polynomials. We study the resultant in the context of mixed multilinear polynomial systems, that is multilinear systems with polynomials having different supports, on which determinantal formulas were not known. We construct determinantal formulas for two kind of multilinear systems related to the Multiparameter Eigenvalue Problem (MEP): first, when the polynomials agree in all but one block of variables; second, when the polynomials are bilinear with different supports, related to a bipartite graph. We use the Weyman complex to construct Koszul-type determinantal formulas that generalize Sylvester-type formulas. We can use the matrices associated to these formulas to solve square systems without computing the resultant. The combination of the resultant matrices with the eigenvalue and eigenvector criterion for polynomial systems leads to a new approach for solving MEP.

math.AC↗

arXiv2021

Geometric algorithms for sampling the flux space of metabolic networks

Systems Biology is a fundamental field and paradigm that introduces a new era in Biology. The crux of its functionality and usefulness relies on metabolic networks that model the reactions occurring inside an organism and provide the means to understand the underlying mechanisms that govern biological systems. Even more, metabolic networks have a broader impact that ranges from resolution of ecosystems to personalized medicine.The analysis of metabolic networks is a computational geometry oriented field as one of the main operations they depend on is sampling uniformly points from polytopes; the latter provides a representation of the steady states of the metabolic networks. However, the polytopes that result from biological data are of very high dimension (to the order of thousands) and in most, if not all, the cases are considerably skinny. Therefore, to perform uniform random sampling efficiently in this setting, we need a novel algorithmic and computational framework specially tailored for the properties of metabolic networks.We present a complete software framework to handle sampling in metabolic networks. Its backbone is a Multiphase Monte Carlo Sampling (MMCS) algorithm that unifies rounding and sampling in one pass, obtaining both upon termination. It exploits an improved variant of the Billiard Walk that enjoys faster arithmetic complexity per step. We demonstrate the efficiency of our approach by performing extensive experiments on various metabolic networks. Notably, sampling on the most complicated human metabolic network accessible today, Recon3D, corresponding to a polytope of dimension 5 335 took less than 30 hours. To our knowledge, that is out of reach for existing software.

q-bio.QM↗

arXiv2020

Efficient Sampling from Feasible Sets of SDPs and Volume Approximation

We present algorithmic, complexity, and implementation results on the problem of sampling points from a spectrahedron, that is the feasible region of a semidefinite program. Our main tool is geometric random walks. We analyze the arithmetic and bit complexity of certain primitive geometric operations that are based on the algebraic properties of spectrahedra and the polynomial eigenvalue problem. This study leads to the implementation of a broad collection of random walks for sampling from spectrahedra that experimentally show faster mixing times than methods currently employed either in theoretical studies or in applications, including the popular family of Hit-and-Run walks. The different random walks offer a variety of advantages , thus allowing us to efficiently sample from general probability distributions, for example the family of log-concave distributions which arise in numerous applications. We focus on two major applications of independent interest: (i) approximate the volume of a spectrahedron, and (ii) compute the expectation of functions coming from robust optimal control. We exploit efficient linear algebra algorithms and implementations to address the aforemen-tioned computations in very high dimension. In particular, we provide a C++ open source implementation of our methods that scales efficiently, for the first time, up to dimension 200. We illustrate its efficiency on various data sets.

cs.CG↗

arXiv2019

Computing the topology of a planar or space hyperelliptic curve

We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is easy to compute, under a birational mapping of the plane or the space. We report on a {\tt Maple} implementation of these algorithms, and present several examples. Complexity and certification issues are also discussed.

math.GT↗

arXiv2019

A nearly optimal algorithm to decompose binary forms

Symmetric tensor decomposition is an important problem with applications in several areas for example signal processing, statistics, data analysis and computational neuroscience. It is equivalent to Waring's problem for homogeneous polynomials, that is to write a homogeneous polynomial in n variables of degree D as a sum of D-th powers of linear forms, using the minimal number of summands. This minimal number is called the rank of the polynomial/tensor. We focus on decomposing binary forms, a problem that corresponds to the decomposition of symmetric tensors of dimension 2 and order D. Under this formulation, the problem finds its roots in invariant theory where the decompositions are known as canonical forms. In this context many different algorithms were proposed. We introduce a superfast algorithm that improves the previous approaches with results from structured linear algebra. It achieves a softly linear arithmetic complexity bound. To the best of our knowledge, the previously known algorithms have at least quadratic complexity bounds. Our algorithm computes a symbolic decomposition in $O(M(D) log(D))$ arithmetic operations, where $M(D)$ is the complexity of multiplying two polynomials of degree D. It is deterministic when the decomposition is unique. When the decomposition is not unique, our algorithm is randomized. We present a Monte Carlo version of it and we show how to modify it to a Las Vegas one, within the same complexity. From the symbolic decomposition, we approximate the terms of the decomposition with an error of $2^{--$ε$}$ , in $O(D log^2(D) (log^2(D) + log($ε$)))$ arithmetic operations. We use results from Kaltofen and Yagati (1989) to bound the size of the representation of the coefficients involved in the decomposition and we bound the algebraic degree of the problem by min(rank, D -- rank + 1). We show that this bound can be tight. When the input polynomial has integer coefficients, our algorithm performs, up to poly-logarithmic factors, $O\_{bit} (D{\ell} + D^4 + D^3 $τ$)$ bit operations, where $$τ$$ is the maximum bitsize of the coefficients and $2^{--{\ell}}$ is the relative error of the terms in the decomposition.

cs.SC↗

arXiv2019

Gr{ö}bner Basis over Semigroup Algebras: Algorithms and Applications for Sparse Polynomial Systems

Gr{ö}bner bases is one the most powerful tools in algorithmic non-linear algebra. Their computation is an intrinsically hard problem with a complexity at least single exponential in the number of variables. However, in most of the cases, the polynomial systems coming from applications have some kind of structure. For example , several problems in computer-aided design, robotics, vision, biology , kinematics, cryptography, and optimization involve sparse systems where the input polynomials have a few non-zero terms. Our approach to exploit sparsity is to embed the systems in a semigroup algebra and to compute Gr{ö}bner bases over this algebra. Up to now, the algorithms that follow this approach benefit from the sparsity only in the case where all the polynomials have the same sparsity structure, that is the same Newton polytope. We introduce the first algorithm that overcomes this restriction. Under regularity assumptions, it performs no redundant computations. Further, we extend this algorithm to compute Gr{ö}bner basis in the standard algebra and solve sparse polynomials systems over the torus $(C*)^n$. The complexity of the algorithm depends on the Newton polytopes.

cs.SC↗

arXiv2018

On the maximal number of real embeddings of minimally rigid graphs in $\mathbb{R}^2$, $\mathbb{R}^3$ and $S^2$

Rigidity theory studies the properties of graphs that can have rigid embeddings in a euclidean space $\mathbb{R}^d$ or on a sphere and which in addition satisfy certain edge length constraints. One of the major open problems in this field is to determine lower and upper bounds on the number of realizations with respect to a given number of vertices. This problem is closely related to the classification of rigid graphs according to their maximal number of real embeddings. In this paper, we are interested in finding edge lengths that can maximize the number of real embeddings of minimally rigid graphs in the plane, space, and on the sphere. We use algebraic formulations to provide upper bounds. To find values of the parameters that lead to graphs with a large number of real realizations, possibly attaining the (algebraic) upper bounds, we use some standard heuristics and we also develop a new method inspired by coupler curves. We apply this new method to obtain embeddings in $\mathbb{R}^3$. One of its main novelties is that it allows us to sample efficiently from a larger number of parameters by selecting only a subset of them at each iteration. Our results include a full classification of the 7-vertex graphs according to their maximal numbers of real embeddings in the cases of the embeddings in $\mathbb{R}^2$ and $\mathbb{R}^3$, while in the case of $S^2$ we achieve this classification for all 6-vertex graphs. Additionally, by increasing the number of embeddings of selected graphs, we improve the previously known asymptotic lower bound on the maximum number of realizations. The methods and the results concerning the spatial embeddings are part of the proceedings of ISSAC 2018 (Bartzos et al, 2018).

math.AG↗