arXiv subjects

Jonathan Hauenstein

Publications and source records attributed to Jonathan Hauenstein.

7 recordsLinked to original sources

arXiv2023

Multi-graded Macaulay Dual Spaces

We describe an algorithm for computing Macaulay dual spaces for multi-graded ideals. For homogeneous ideals, the natural grading is inherited by the Macaulay dual space which has been leveraged to develop algorithms to compute the Macaulay dual space in each homogeneous degree. Our main theoretical result extends this idea to multi-graded Macaulay dual spaces inherited from multi-graded ideals. This natural duality allows ideal operations to be translated from homogeneous ideals to their corresponding operations on the multi-graded Macaulay dual spaces. In particular, we describe a linear operator with a right inverse for computing quotients by a multi-graded polynomial. By using a total ordering on the homogeneous components of the Macaulay dual space, we also describe how to recursively construct a basis for each component. Several examples are included to demonstrate this new approach.

math.AC↗

arXiv2019

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↗

arXiv2018

Certifying reality of projections

Computational tools in numerical algebraic geometry can be used to numerically approximate solutions to a system of polynomial equations. If the system is well-constrained (i.e., square), Newton's method is locally quadratically convergent near each nonsingular solution. In such cases, Smale's alpha theory can be used to certify that a given point is in the quadratic convergence basin of some solution. This was extended to certifiably determine the reality of the corresponding solution when the polynomial system is real. Using the theory of Newton-invariant sets, we certifiably decide the reality of projections of solutions. We apply this method to certifiably count the number of real and totally real tritangent planes for instances of curves of genus 4.

math.AG↗

arXiv2015

Complexity of linear circuits and geometry

We use algebraic geometry to study matrix rigidity, and more generally, the complexity of computing a matrix-vector product, continuing a study initiated by Kumar, et. al. We (i) exhibit many non-obvious equations testing for (border) rigidity, (ii) compute degrees of varieties associated to rigidity, (iii) describe algebraic varieties associated to families of matrices that are expected to have super-linear rigidity, and (iv) prove results about the ideals and degrees of cones that are of interest in their own right.

cs.CC↗

arXiv2013

Maximum Likelihood for Matrices with Rank Constraints

Maximum likelihood estimation is a fundamental optimization problem in statistics. We study this problem on manifolds of matrices with bounded rank. These represent mixtures of distributions of two independent discrete random variables. We determine the maximum likelihood degree for a range of determinantal varieties, and we apply numerical algebraic geometry to compute all critical points of their likelihood functions. This led to the discovery of maximum likelihood duality between matrices of complementary ranks, a result proved subsequently by Draisma and Rodriguez.

math.AG↗

arXiv2012

Numerical Analyses on Moduli Space of Vacua

We propose a new computational method to understand the vacuum moduli space of (supersymmetric) field theories. By combining numerical algebraic geometry (NAG) and elimination theory, we develop a powerful, efficient, and parallelizable algorithm to extract important information such as the dimension, branch structure, Hilbert series and subsequent operator counting, as well as variation according to coupling constants and mass parameters. We illustrate this method on a host of examples from gauge theory, string theory, and algebraic geometry.

hep-th↗

arXiv2011

Algebraic boundaries of Hilbert's SOS cones

We study the geometry underlying the difference between non-negative polynomials and sums of squares. The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether-Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized respectively by the Severi variety of plane rational sextics and by the variety of quartic symmetroids.

math.AG↗