arXiv subjects

Jose Israel Rodriguez

Publications and source records attributed to Jose Israel Rodriguez.

At least 37 records · Page 2Linked to original sources

arXiv2022

Logarithmic cotangent bundles, Chern-Mather classes, and the Huh-Sturmfels Involution conjecture

Using compactifications in the logarithmic cotangent bundle, we obtain a formula for the Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement. This generalizes earlier results of Aluffi and Wu-Zhou. The first application of our formula is a geometric description of Chern-Mather classes of an arbitrary very affine variety, generalizing earlier results of Huh which held under the smooth and schon assumptions. As the second application, we confirm an involution formula relating sectional maximum likelihood (ML) degrees and ML bidegrees, which was conjectured by Huh and Sturmfels in 2013.

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arXiv2021

Solving Parameterized Polynomial Systems with Decomposable Projections

The Galois group of a parameterized polynomial system of equations encodes the structure of the solutions. This monodromy group acts on the set of solutions for a general set of parameters, that is, on the fiber of a projection from the incidence variety of parameters and solutions onto the space of parameters. When this projection is decomposable, the Galois group is imprimitive, and we show that the structure can be exploited for computational improvements. Furthermore, we develop a new algorithm for solving these systems based on a suitable trace test. We illustrate our method on examples in statistics, kinematics, and benchmark problems in computational algebra. In particular, we resolve a conjecture on the number of solutions of the moment system associated to a mixture of Gaussian distributions.

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arXiv2020

Fiber product homotopy method for multiparameter eigenvalue problems

We develop a new homotopy method for solving multiparameter eigenvalue problems (MEPs) called the fiber product homotopy method. For a $k$-parameter eigenvalue problem with matrices of sizes $n_1,\dots ,n_k = O(n)$, fiber product homotopy method requires deformation of $O(1)$ linear equations, while existing homotopy methods for MEPs require $O(n)$ nonlinear equations. We show that the fiber product homotopy method theoretically finds all eigenpairs of an MEP with probability one. It is especially well-suited for dimension-deficient singular MEPs, a weakness of all other existing methods, as the fiber product homotopy method is provably convergent with probability one for such problems as well, a fact borne out by numerical experiments. More generally, our numerical experiments indicate that the fiber product homotopy method significantly outperforms the standard Delta method in terms of accuracy, with consistent backward errors on the order of $10^{-16}$, even for dimension-deficient singular problems, and without any use of extended precision. In terms of speed, it significantly outperforms previous homotopy-based methods on all problems and outperforms the Delta method on larger problems, and is also highly parallelizable. We show that the fiber product MEP that we solve in the fiber product homotopy method, although mathematically equivalent to a standard MEP, is typically a much better conditioned problem.

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arXiv2020

Solving Decomposable Sparse Systems

Amendola et al. proposed a method for solving systems of polynomial equations lying in a family which exploits a recursive decomposition into smaller systems. A family of systems admits such a decomposition if and only if the corresponding Galois group is imprimitive. When the Galois group is imprimitive we consider the problem of computing an explicit decomposition. A consequence of Esterov's classification of sparse polynomial systems with imprimitive Galois groups is that this decomposition is obtained by inspection. This leads to a recursive algorithm to solve decomposable sparse systems, which we present and give evidence for its efficiency.

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arXiv2020

Multiregeneration for polynomial system solving

We demonstrate our implementation of a continuation method as described in \cite{HR2015} for solving polynomials systems. Given a sequence of (multi)homogeneous polynomials, the software "multiregeneration" outputs the respective (multi)degree in a wide range of cases and partial multidegree in all others. We use Python for the file processing, while Bertini is needed for the continuation. Moreover, parallelization options and several strategies for solving structured polynomial systems are available.

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arXiv2020

Decomposable sparse polynomial systems

The Macaulay2 package DecomposableSparseSystems implements methods for studying and numerically solving decomposable sparse polynomial systems. We describe the structure of decomposable sparse systems and explain how the methods in this package may be used to exploit this structure, with examples.

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arXiv2020

A numerical toolkit for multiprojective varieties

A numerical description of an algebraic subvariety of projective space is given by a general linear section, called a witness set. For a subvariety of a product of projective spaces (a multiprojective variety), the corresponding numerical description is given by a witness collection, whose structure is more involved. We build on recent work to develop a toolkit for the numerical manipulation of multiprojective varieties that operates on witness collections, and use this toolkit in an algorithm for numerical irreducible decomposition of multiprojective varieties. The toolkit and decomposition algorithm are illustrated throughout in a series of examples.

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arXiv2020

A Morse theoretic approach to non-isolated singularities and applications to optimization

Let $X$ be a complex affine variety in $\mathbb{C}^N$, and let $f:\mathbb{C}^N\to \mathbb{C}$ be a polynomial function whose restriction to $X$ is nonconstant. For $g:\mathbb{C}^N \to \mathbb{C}$ a general linear function, we study the limiting behavior of the critical points of the one-parameter family of $f_t: =f-tg$ as $t\to 0$. Our main result gives an expression of this limit in terms of critical sets of the restrictions of $g$ to the singular strata of $(X,f)$. We apply this result in the context of optimization problems. For example, we consider nearest point problems (e.g., Euclidean distance degrees) for affine varieties and a possibly nongeneric data point.

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arXiv2020

The algebraic matroid of the funtf variety

A finite unit norm tight frame is a collection of $r$ vectors in $\mathbb{R}^n$ that generalizes the notion of orthonormal bases. The affine finite unit norm tight frame variety is the Zariski closure of the set of finite unit norm tight frames. Determining the fiber of a projection of this variety onto a set of coordinates is called the algebraic finite unit norm tight frame completion problem. Our techniques involve the algebraic matroid of an algebraic variety, which encodes the dimensions of fibers of coordinate projections. This work characterizes the bases of the algebraic matroid underlying the variety of finite unit norm tight frames in $\mathbb{R}^3$. Partial results towards similar characterizations for finite unit norm tight frames in $\mathbb{R}^n$ with $n \ge 4$ are also given. We provide a method to bound the degree of the projections based off of combinatorial~data.

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arXiv2019

Defect of Euclidean distance degree

Two well studied invariants of a complex projective variety are the unit Euclidean distance degree and the generic Euclidean distance degree. These numbers give a measure of the algebraic complexity for "nearest" point problems of the algebraic variety. It is well known that the latter is an upper bound for the former. While this bound may be tight, many varieties appearing in optimization, engineering, statistics, and data science, have a significant gap between these two numbers. We call this difference the defect of the ED degree of an algebraic variety. In this paper we compute this defect by classical techniques in Singularity Theory, thereby deriving a new method for computing ED degrees of smooth complex projective varieties.

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arXiv2019

Euclidean distance degree of projective varieties

We give a positive answer to a conjecture of Aluffi-Harris on the computation of the Euclidean distance degree of a possibly singular projective variety in terms of the local Euler obstruction function.

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arXiv2018

Euclidean distance degree of the multiview variety

The Euclidean distance degree of an algebraic variety is a well-studied topic in applied algebra and geometry. It has direct applications in geometric modeling, computer vision, and statistics. We use non-proper Morse theory to give a topological interpretation of the Euclidean distance degree of an affine variety in terms of Euler characteristics. As a concrete application, we solve the open problem in computer vision of determining the Euclidean distance degree of the affine multiview variety.

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arXiv2018

Solving the likelihood equations to compute Euler obstruction functions

Macpherson defined Chern-Schwartz-Macpherson (CSM) classes by introducing the (local) Euler obstruction function, which is an integer valued function on the variety that is constant on each stratum of a Whitney stratification of an algebraic variety. By understanding the Euler obstruction function, one gains insights about a singular algebraic variety. It was recently shown by the author and B. Wang, how to compute these functions using maximum likelihood degrees. This paper discusses a symbolic and a numerical implementation of algorithms to compute the Euler obstruction at a point. Macaulay2 and Bertini are used in the implementations.

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arXiv2018

Computing Euler obstruction functions using maximum likelihood degrees

We give a numerical algorithm computing Euler obstruction functions using maximum likelihood degrees. The maximum likelihood degree is a well-studied property of a variety in algebraic statistics and computational algebraic geometry. In this article we use this degree to give a new way to compute Euler obstruction functions. We define the maximum likelihood obstruction function and show how it coincides with the Euler obstruction function. With this insight, we are able to bring new tools of computational algebraic geometry to study Euler obstruction functions.

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arXiv2017

Numerical computation of braid groups

In this article, we give a numerical algorithm to compute braid groups of curves, hyperplane arrangements, and parameterized system of polynomial equations. Our main result is an algorithm that determines the cross-locus and the generators of the braid group.

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arXiv2017

The Maximum Likelihood Degree of Toric Varieties

We study the maximum likelihood degree (ML degree) of toric varieties, known as discrete exponential models in statistics. By introducing scaling coefficients to the monomial parameterization of the toric variety, one can change the ML degree. We show that the ML degree is equal to the degree of the toric variety for generic scalings, while it drops if and only if the scaling vector is in the locus of the principal $A$-determinant. We also illustrate how to compute the ML estimate of a toric variety numerically via homotopy continuation from a scaled toric variety with low ML degree. Throughout, we include examples motivated by algebraic geometry and statistics. We compute the ML degree of rational normal scrolls and a large class of Veronese-type varieties. In addition, we investigate the ML degree of scaled Segre varieties, hierarchical loglinear models, and graphical models.

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arXiv2017

Accurate Solutions of Polynomial Eigenvalue Problems

Quadratic eigenvalue problems (QEP) and more generally polynomial eigenvalue problems (PEP) are among the most common types of nonlinear eigenvalue problems. Both problems, especially the QEP, have extensive applications. A typical approach to solve QEP and PEP is to use a linearization method to reformulate the problem as a higher dimensional linear eigenvalue problem. In this article, we use homotopy continuation to solve these nonlinear eigenvalue problems without passing to higher dimensions. Our main contribution is to show that our method produces substantially more accurate results, and finds all eigenvalues with a certificate of correctness via Smale's $α$-theory. To explain the superior accuracy, we show that the nonlinear eigenvalue problem we solve is better conditioned than its reformulated linear eigenvalue problem, and our homotopy continuation algorithm is more stable than QZ algorithm - theoretical findings that are borne out by our numerical experiments. Our studies provide yet another illustration of the dictum in numerical analysis that, for reasons of conditioning and stability, it is sometimes better to solve a nonlinear problem directly even when it could be transformed into a linear problem with the same solution mathematically.

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arXiv2017

Trace test

The trace test in numerical algebraic geometry verifies the completeness of a witness set of an irreducible variety in affine or projective space. We give a brief derivation of the trace test and then consider it for subvarieties of products of projective spaces using multihomogeneous witness sets. We show how a dimension reduction leads to a practical trace test in this case involving a curve in a low-dimensional affine space.

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