arXiv subjects

Jonathan D. Hauenstein

Publications and source records attributed to Jonathan D. Hauenstein.

At least 37 records · Page 2Linked to original sources

arXiv2022

Output Mode Switching for Parallel Five-bar Manipulators Using a Graph-based Path Planner

The configuration manifolds of parallel manipulators exhibit more nonlinearity than serial manipulators. Qualitatively, they can be seen to possess extra folds. By projecting such manifolds onto spaces of engineering relevance, such as an output workspace or an input actuator space, these folds cast edges that exhibit nonsmooth behavior. For example, inside the global workspace bounds of a five-bar linkage appear several local workspace bounds that only constrain certain output modes of the mechanism. The presence of such boundaries, which manifest in both input and output projections, serve as a source of confusion when these projections are studied exclusively instead of the configuration manifold itself. Particularly, the design of nonsymmetric parallel manipulators has been confounded by the presence of exotic projections in their input and output spaces. In this paper, we represent the configuration space with a radius graph, then weight each edge by solving an optimization problem using homotopy continuation to quantify transmission quality. We then employ a graph path planner to approximate geodesics between configuration points that avoid regions of low transmission quality. Our methodology automatically generates paths capable of transitioning between non-neighboring output modes, a motion which involves osculating multiple workspace boundaries (local, global, or both). We apply our technique to two nonsymmetric five-bar examples that demonstrate how transmission properties and other characteristics of the workspace can be selected by switching output modes.

cs.RO↗

arXiv2022

Computing geometric feature sizes for algebraic manifolds

We introduce numerical algebraic geometry methods for computing lower bounds on the reach, local feature size, and the weak feature size of the real part of an equidimensional and smooth algebraic variety using the variety's defining polynomials as input. For the weak feature size, we also show that non-quadratic complete intersections generically have finitely many geometric bottlenecks, and describe how to compute the weak feature size directly rather than a lower bound in this case. In all other cases, we describe additional computations that can be used to determine feature size values rather than lower bounds. We also present homology inference experiments that combine persistent homology computations with implemented versions of our feature size algorithms, both with globally dense samples and samples that are adaptively dense with respect to the local feature size.

math.AG↗

arXiv2022

Machine learning the real discriminant locus

Parameterized systems of polynomial equations arise in many applications in science and engineering with the real solutions describing, for example, equilibria of a dynamical system, linkages satisfying design constraints, and scene reconstruction in computer vision. Since different parameter values can have a different number of real solutions, the parameter space is decomposed into regions whose boundary forms the real discriminant locus. This article views locating the real discriminant locus as a supervised classification problem in machine learning where the goal is to determine classification boundaries over the parameter space, with the classes being the number of real solutions. For multidimensional parameter spaces, this article presents a novel sampling method which carefully samples the parameter space. At each sample point, homotopy continuation is used to obtain the number of real solutions to the corresponding polynomial system. Machine learning techniques including nearest neighbor and deep learning are used to efficiently approximate the real discriminant locus. One application of having learned the real discriminant locus is to develop a real homotopy method that only tracks the real solution paths unlike traditional methods which track all~complex~solution~paths. Examples show that the proposed approach can efficiently approximate complicated solution boundaries such as those arising from the equilibria of the Kuramoto model.

stat.ML↗

arXiv2022

AdaAnn: Adaptive Annealing Scheduler for Probability Density Approximation

Approximating probability distributions can be a challenging task, particularly when they are supported over regions of high geometrical complexity or exhibit multiple modes. Annealing can be used to facilitate this task which is often combined with constant a priori selected increments in inverse temperature. However, using constant increments limit the computational efficiency due to the inability to adapt to situations where smooth changes in the annealed density could be handled equally well with larger increments. We introduce AdaAnn, an adaptive annealing scheduler that automatically adjusts the temperature increments based on the expected change in the Kullback-Leibler divergence between two distributions with a sufficiently close annealing temperature. AdaAnn is easy to implement and can be integrated into existing sampling approaches such as normalizing flows for variational inference and Markov chain Monte Carlo. We demonstrate the computational efficiency of the AdaAnn scheduler for variational inference with normalizing flows on a number of examples, including density approximation and parameter estimation for dynamical systems.

stat.CO↗

arXiv2021

On computing the nonlinearity interval in parametric semidefinite optimization

This paper revisits the parametric analysis of semidefinite optimization problems with respect to the perturbation of the objective function along a fixed direction. We review the notions of invariancy set, nonlinearity interval, and transition point of the optimal partition, and we investigate their characterizations. We show that the set of transition points is finite and the continuity of the optimal set mapping, on the basis of Painlevé-Kuratowski set convergence, might fail on a nonlinearity interval. Under a local nonsingularity condition, we then develop a methodology, stemming from numerical algebraic geometry, to efficiently compute nonlinearity intervals and transition points of the optimal partition. Finally, we support the theoretical results by applying our procedure to some numerical examples.

math.OC↗

arXiv2021

On the equations defining some Hilbert schemes

We work out details of the extrinsic geometry for two Hilbert schemes of some contemporary interest: the Hilbert scheme of two points on the projective plane and the dense open set parametrizing non-planar clusters in the punctual Hilbert scheme of clusters of length four on affine three-space with support at the origin. We find explicit equations in natural projective, respectively affine embeddings for these spaces. In particular, we answer a question of Bernd Sturmfels who asked for a description of the latter space that is amenable to further computations. While the explicit equations we find are controlled in a precise way by the representation theory of SL_3, our arguments also rely on computer algebra.

math.AG↗

arXiv2021

Certified evaluations of Hölder continuous functions at roots of polynomials

Various methods can obtain certified estimates for roots of polynomials. Many applications in science and engineering additionally utilize the value of functions evaluated at roots. For example, critical values are obtained by evaluating an objective function at critical points. For analytic evaluation functions, Newton's method naturally applies to yield certified estimates. These estimates no longer apply, however, for Hölder continuous functions, which are a generalization of Lipschitz continuous functions where continuous derivatives need not exist. This work develops and analyzes an alternative approach for certified estimates of evaluating locally Hölder continuous functions at roots of polynomials. An implementation of the method in Maple demonstrates efficacy and efficiency.

cs.SC↗

arXiv2020

Using monodromy to statistically estimate the number of solutions

Synthesis problems for linkages in kinematics often yield large structured parameterized polynomial systems which generically have far fewer solutions than traditional upper bounds would suggest. This paper describes statistical models for estimating the generic number of solutions of such parameterized polynomial systems. The new approach extends previous work on success ratios of parameter homotopies to using monodromy loops as well as the addition of a trace test that provides a stopping criterion for validating that all solutions have been found. Several examples are presented demonstrating the method including Watt I six-bar motion generation problems.

cs.RO↗

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.

math.AG↗

arXiv2019

Probabilistic Saturations and Alt's Problem

Alt's problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial equations which was first solved numerically using homotopy continuation by Wampler, Morgan, and Sommese in 1992. Since there is still not a proof that all solutions were obtained, we consider upper bounds for Alt's problem by counting the number of solutions outside of the base locus to a system arising as the general linear combination of polynomials. In particular, we derive effective symbolic and numeric methods for studying such systems using probabilistic saturations that can be employed using both finite fields and floating-point computations. We give bounds on the size of finite field required to achieve a desired level of certainty. These methods can also be applied to many other problems where similar systems arise such as computing the volumes of Newton-Okounkov bodies and computing intersection theoretic invariants including Euler characteristics, Chern classes, and Segre classes.

math.AG↗

arXiv2019

Real monodromy action

The monodromy group is an invariant for parameterized systems of polynomial equations that encodes structure of the solutions over the parameter space. Since the structure of real solutions over real parameter spaces are of interest in many applications, real monodromy action is investigated here. A naive extension of monodromy action from the complex numbers to the real numbers is shown to be very restrictive. Therefore, we define a real monodromy structure which need not be a group but contains tiered characteristics about the real solutions. This real monodromy structure is applied to an example in kinematics which summarizes all the ways performing loops parameterized by leg lengths can cause a mechanism to change poses.

math.AG↗

arXiv2018

Sampling real algebraic varieties for topological data analysis

Topological data analysis (TDA) provides a growing body of tools for computing geometric and topological information about spaces from a finite sample of points. We present a new adaptive algorithm for finding provably dense samples of points on real algebraic varieties given a set of defining polynomials. The algorithm utilizes methods from numerical algebraic geometry to give formal guarantees about the density of the sampling and it also employs geometric heuristics to reduce the size of the sample. As TDA methods consume significant computational resources that scale poorly in the number of sample points, our sampling minimization makes applying TDA methods more feasible. We provide a software package that implements the algorithm and also demonstrate the implementation with several examples.

math.AT↗

arXiv2018

The loss surface of deep linear networks viewed through the algebraic geometry lens

By using the viewpoint of modern computational algebraic geometry, we explore properties of the optimization landscapes of the deep linear neural network models. After clarifying on the various definitions of "flat" minima, we show that the geometrically flat minima, which are merely artifacts of residual continuous symmetries of the deep linear networks, can be straightforwardly removed by a generalized $L_2$ regularization. Then, we establish upper bounds on the number of isolated stationary points of these networks with the help of algebraic geometry. Using these upper bounds and utilizing a numerical algebraic geometry method, we find all stationary points of modest depth and matrix size. We show that in the presence of the non-zero regularization, deep linear networks indeed possess local minima which are not the global minima. Our computational results clarify certain aspects of the loss surfaces of deep linear networks and provide novel insights.

stat.ML↗

arXiv2018

Singular value decomposition of complexes

Singular value decompositions of matrices are widely used in numerical linear algebra with many applications. In this paper, we extend the notion of singular value decompositions to finite complexes of real vector spaces. We provide two methods to compute them and present several applications.

math.NA↗

arXiv2018

Solving determinantal systems using homotopy techniques

Let $\K$ be a field of characteristic zero and $\Kbar$ be an algebraic closure of $\K$. Consider a sequence of polynomials$G=(g\_1,\dots,g\_s)$ in $\K[X\_1,\dots,X\_n]$, a polynomial matrix $\F=[f\_{i,j}] \in \K[X\_1,\dots,X\_n]^{p \times q}$, with $p \leq q$,and the algebraic set $V\_p(F, G)$ of points in $\KKbar$ at which all polynomials in $\G$ and all $p$-minors of $\F$vanish. Such polynomial systems appear naturally in e.g. polynomial optimization, computational geometry.We provide bounds on the number of isolated points in $V\_p(F, G)$ depending on the maxima of the degrees in rows (resp. columns) of $\F$. Next, we design homotopy algorithms for computing those points. These algorithms take advantage of the determinantal structure of the system defining $V\_p(F, G)$. In particular, the algorithms run in time that is polynomial in the bound on the number of isolated points.

cs.SC↗

arXiv2018

Optimal Configurations in Coverage Control with Polynomial Costs

We revisit the static coverage control problem for placement of vehicles with simple motion on the real line, under the assumption that the cost is a polynomial function of the locations of the vehicles. The main contribution of this paper is to demonstrate the use of tools from numerical algebraic geometry, in particular, a numerical polynomial homotopy continuation method that guarantees to find all solutions of polynomial equations, in order to characterize the \emph{global minima} for the coverage control problem. The results are then compared against a classic distributed approach involving the use of Lloyd descent, which is known to converge only to a local minimum under certain technical conditions.

eess.SY↗

arXiv2017

Locating and counting equilibria of the Kuramoto model with rank one coupling

The Kuramoto model describes synchronization behavior among coupled oscillators and enjoys successful application in a wide variety of fields. Many of these applications seek phase-coherent solutions, i.e., equilibria of the model. Historically, research has focused on situations where the number of oscillators, $n$, is extremely large and can be treated as being infinite. More recently, however, applications have arisen in areas such as electrical engineering with more modest values of $n$. For these, the equilibria can be located by finding the real solutions of a system of polynomial equations utilizing techniques from algebraic geometry. However, typical methods for solving such systems locate all complex solutions even though only the real solutions give equilibria. In this paper, we present an algorithm to locate only the real solutions of the model, thereby shortening computation time by several orders of magnitude in certain situations. This is accomplished by choosing specific equilibria representatives and the consequent algebraic decoupling of the system. The correctness of the algorithm (that it finds only and all the equilibria) is proved rigorously. Additionally, the algorithm can be implemented using interval methods so that the equilibria can be approximated up to any given precision without significantly more computational effort. We also compare this solving approach to other computational algebraic geometric methods. Furthermore, analyzing this approach allows us to prove, asymptotically, that the maximum number of equilibria grows at the same rate as the number of complex solutions of a corresponding polynomial system. Finally, we conjecture an upper bound on the maximum number of equilibria for any number of oscillators which generalizes the known cases and is obtained on a range of explicitly provided natural frequencies.

math.OC↗

arXiv2017

Adaptive strategies for solving parameterized systems using homotopy continuation

Three aspects of applying homotopy continuation, which is commonly used to solve parameterized systems of polynomial equations, are investigated. First, for parameterized systems which are homogeneous, we investigate options for performing computations on an adaptively chosen affine coordinate patch. Second, for parameterized systems which are overdetermined, we investigate options for adaptively selecting a well-constrained subsystem to restore numerical stability. Finally, since one is typically interested in only computing real solutions for parameterized problems which arise from applications, we investigate a scheme for heuristically identifying solution paths which appear to be ending at nonreal solutions and truncating them. We demonstrate these three aspects on two problems arising in computer vision.

math.NA↗