Research
Applied Algebra
Algebraic Statistics
Nonlinear Algebra
Research Aspects
My research is about studying problems in Applied Algebraic Geometry motivated by Algebraic Statistics.
Applied Algebraic Geometry is an inclusive field involving industry, scientists, and mathematicians. Our main conference SIAM AG happens every two years and the top journal is SIAM Applied Algebra and Geometry (SIAGA). Students who work with me are encouraged to participate in this community. One way to do this stems from the global lockdown in 2020: I coorganized our community’s monthly webinar, SIAM SAGA, which has recordings archived. Another way is to attend our Applied Algebra Seminar or to join the SIAM Algebraic Geometry Activity Group (free memberships are often available for students).
Algebraic Statistics is a branch of mathematical statistics that focuses on the use of algebraic, geometric and combinatorial methods in statistics. In Fall 2023, I will coorganize a semester long program at the new NSF institute IMSI (Chicago) titled Algebraic Statistics and Our Changing World: New Methods for New Challenges (Send me an email if interested). Also, I have organized special sessions with Algebraic Statistics Themes at the AMS Sectional meetings, the SIAM Annual Meeting, and Joint Statistics Meeting. Many of my students begin working with me on problems in this area and branch out into others.
More specifically, my research involves three key aspects that intertwine and motivate one another. The first aspect is to determine algebraic degrees of optimization problems. Many optimization problems can be phrased as solving a polynomial system whose roots correspond to critical points. Several of my results are in the context where the objective function measures Euclidean distance or likelihood. The number of critical points the associated objective on a given model is an invariant called the Euclidean Distance (ED) degree and Maximum Likelihood degree respectively. Using a combination of tools including numerical nonlinear algebra, symbolic computation, and singularity theory, I compute these invariants. Some applications of this work is to view the ED degree as an Euler characteristic and then use this point of view to prove a formula for the number of critical points for the n-view triangulation problem.
To solve global optimization problems, the second aspect of my research is to develop new techniques in numerical algebraic geometry. For example, with collaborators I developed a fiber product homotopy method to study multiparameter eigenvalue problems (MEP), I introduced new algorithms for solving decomposable sparse polynomial systems, and created a toolkit for studying multiprojective varieties. Homotopy continuation underlies these methods, and one strength of algebraic geometry comes from the notion of degree. Knowing the degree of an optimization problem allows the practitioner to make provable guarantees about finding a global optimum.
This leads to the third aspect of my research. I put the theoretical ideas into practice. In a recent collaboration, we do density estimation for n-dimensional Gaussian mixtures using the tools of algebraic geometry. These are large scale problems that have hundreds of thousands of unknown parameters that can be accurately recovered using our method. More broadly, I look to apply my techniques in other applications like kinematics, data science, and computer vision. This is in addition to building community by supplying implementations.
If you are a student who is interested in any of these aspects, then I am happy to chat with you.
For Undergraduates
I am available for directed reading courses on Ideal, Varieties, and Algorithms (introduction to computational algebraic geometry), Algebraic Statistics, Solving polynomial systems, and related references.
Expectations for directed readings: I expect to meet once to twice a week for ~25 minutes. During this time you are encouraged to tell me what you found interesting and to present a worked out example. This is followed up with notes in overleaf. If you like, I am also happy to give feedback on your notes.
Undergraduate research: I strongly encourage students to be familiar with Ideal, Varieties, and Algorithms OR to have worked through this crash course with HomotopyContinuation.jl before committing to a research project. If you need help getting started it might be better to do a reading course that leads into a project.
In 2020, I ran the Collaborative Undergraduate Research Laboratory (CURL): Applications and outline are found here.
In Spring 2023, with Aviva Englander (Graduate Student) I am mentoring three undergraduates in an MXM project on Algebraic Systems Biology.
Preprints
[**] New directions in algebraic statistics: Three challenges from 2023
(Yulia Alexandr, Miles Bakenhus, Mark Curiel, Sameer K. Deshpande, Elizabeth Gross, Yuqi Gu, Max Hill, Joseph Johnson, Bryson Kagy, Vishesh Karwa, Jiayi Li, Hanbaek Lyu, Sonja Petrović, Jose Israel Rodriguez)
[https://arxiv.org/abs/2402.13961][**] Maximum likelihood estimation for unrooted 3-lead trees: an analytic solution for the CFN model
(Max Hill, Sebastien Roch, Jose Israel Rodriguez)
Biorxiv-2024-578504v1[**] Applications of singularity theory in applied algebraic geometry and algebraic statistics
(Laurentiu Maxim, Jose Israel Rodriguez, Botong Wang)
[http://arxiv.org/abs/2305.19842][**] Estimating Gaussian mixtures using sparse polynomial moment systems
(Julia Lindberg, Carlos Améndola, Jose Israel Rodriguez)
[Arxiv:2106.15675].
We present an algorithm to do parameter recovery for high dimensional Gaussian mixtures that scales linearly in the dimension.[*] Solving parameterized polynomial systems with decomposable projections
(Carlos Améndola, Julia Lindberg, Jose Israel Rodriguez)
[Arxiv:1612.08807].
An early version was presented at MEGA and a recent version disproves a conjecture of Amendola, Faugere and Sturmfels.
We exploit the structure of decomposable projections in problems from statistics, kinematics, and benchmark problems in computational algebra.[*] Numerical computation of braid groups.
(JIR and Botong Wang)
[Arxiv:1711.07947]
We give a numerical algorithm to compute braid groups of curves, hyperplane arrangements, and parameterized system of polynomial equations using homotopy continuation.[*] Accurate Solutions of Polynomial Eigenvalue Problems.
(Yiling You, Jose Israel Rodriguez, Lek-Heng Lim)
[Arxiv:1711.01301]
We use homotopy continuation to solve the polynomial eigenvalue problem. We show that this method produces substantially more accurate results and finds all eigenvalues with a certificate of correctness via Smale's alpha-theory.[*] Bertini for Macaulay2.
(Daniel J. Bates, Elizabeth Gross, Anton Leykin, Jose Israel Rodriguez)
[Arxiv:1310.3297]
Numerical algebraic geometry is the field of computational mathematics concerning the numerical solution of polynomial systems of equations. Bertini, a popular software package for computational applications of this field, includes implementations of a variety of algorithms based on polynomial homotopy continuation. The Macaulay2 package Bertini.m2 provides an interface to Bertini.
Publications🌴
🌴[40] Linear optimization on varieties and Chern-Mather classes
(Laurentiu G. Maxim, Jose Israel Rodriguez, Botong Wang, Lei Wu)🌴 [39] Implementing Real Polyhedral Homotopy
(Kisun Lee, Julia Lindberg, Jose Israel Rodriguez)
🔹Accepted to JSAG [ArXiv:2207.05323].
🔺We implement a real polyhedral homotopy method to compute the real solutions of polynomial system when it is patchworked. .🌴[38] Maximum likelihood degrees, Euclidean distance degrees, and the topology underneath
(Jose Israel Rodriguez) [Extended Abstract]
🔹 Accepted into a Oberwolfach Report
🔺An extended abstract.🌴[38] Invariants of SDP exactness in quadratic programming
(Julia Lindberg, Jose Israel Rodriguez) [ArXiv:2211.05645 ]
🔹 Accepted to JSCo
🔺Consider SDP exact regions with applications to binary programs
🌴[37] u-generation: solving systems of polynomials equation-by-equation
(Timothy Duff, Anton Leykin, Jose Israel Rodriguez) [ArXiv:2206.02869].
🔹 Accepted to Numerical Algorithms.
🔺 We develop a new method that improves the efficiency of equation-by-equation algorithms for solving polynomial systems.
🌴[36] Logarithmic cotangent bundles, Chern-Mather classes, and the Huh-Sturmfels Involution conjecture
(Laurentiu G. Maxim, Jose Israel Rodriguez, Botong Wang, Lei Wu) [Arxiv:2202.00554].
🔹 Accepted to CPAM.
🔺 We prove an involution formula relating sectional maximum likelihood (ML) degrees and ML bidegrees, which was conjectured by Huh and Sturmfels in 2013.
🌴[35] Encountering singularities of a serial robot along continuous paths at high precision
(P. Milenkovic, J. I. Rodriguez, and Z. Wang)
[DOI
🔹Mechanism and Machine Theory.
🔺A new variable step (VS) method rapidly calculates continuous kinematic paths that encounter singularities of a serial robot.
🌴[34] Maximum likelihood degrees of sparse polynomial systems
(Julia Lindberg, Nathan Nicholson, Jose Israel Rodriguez, Zinan Wang)
[Arxiv]
🔹 Accepted to SIAGA.
🔺We prove the maximum likelihood degree of a sparse polynomial system equals the mixed volume of a related Lagrange system of equations.🌴[33] Data loci in algebraic optimization
(Emil Horobet, Jose Israel Rodriguez)
[Arxiv] [DOI]
🔹 To appear in JPAA.
🔺 We introduce methods for computing data loci.🌴[32] Decomposable Sparse Polynomial Systems
(Taylor Brysiewicz, Jose Israel Rodriguez, Frank Sottile, Thomas Yahl)
[Arxiv] [DOI].
🔹J. Softw. Algebra Geom., vol. 11, no. 1, pp. 53–59, 2021.
🔺This article surveys a Macaulay2 implementation for solving sparse decomposable systems.🌴[31] A Morse theoretic approach to non-isolated singularities and applications to optimization
(Laurentiu G. Maxim, Jose Israel Rodriguez, Botong Wang)
[Arxiv] [DOI].
🔹Pure Appl. Algebra, vol. 226,no. 3, p. Paper No. 106865, 2022.
🔺We develop an approach to study positive dimensional critical sets with applications to SDP's and nearest point problems.🌴[30] Solving Decomposable Sparse Systems
(Taylor Brysiewicz, Jose Israel Rodriguez, Frank Sottile, Thomas Yahl)
[Arxiv] [DOI].
🔹Numer. Algorithms, vol. 88, no. 1, pp. 453–474, 2021.
🔺We present a new recursive algorithm for solving sparse decomposable systems.🌴[29] Defect of Euclidean distance degree
(Laurentiu G. Maxim, Jose Israel Rodriguez, Botong Wang)
[Arxiv] [DOI].
🔹 Adv. in Appl. Math. 121 (2020), 102101, 22 pp. 13P25
🔺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. In this paper we compute the difference of these degrees by classical techniques in Singularity Theory, thereby deriving a new method for computing ED degrees of smooth complex projective varieties.🌴[28] Fiber product homotopy method for multiparameter eigenvalue problems
(JIR, Jin-Hong Du, Yiling You & Lek-Heng Lim)
[Arxiv] [DOI].
🔹Numer. Math., vol. 148, no. 4, pp. 853–888, 2021.
🔺We introduce the fiber product homotopy method to solve multiparameter eigenvalue problems. We show that our method is more accurate and faster for large dimensional problems.🌴[27] A numerical toolkit for multiprojective varieties
(J. D. Hauenstein, A. Leykin, JIR, and F. Sottile)
[Arxiv] [DOI].
🔹Math. Comp., vol. 90, no. 327, pp. 413–440, 2021.
🔺We develop a toolkit for studying roots of polynomial systems in multiple variable groups.🌴[26] Euclidean distance degree of projective varieties
(L. G. Maxim, JIR, and B. Wang)
[Arxiv] [DOI].
🔹 Int. Math. Res. Not. IMRN 2021, no. 20, 15788–15802.
🔺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.🌴[25] The algebraic matroid of the funtf variety
(Daniel Irving Bernstein, Cameron Farnsworth, JIR)
[Arxiv] [DOI].
🔹J. Pure Appl. Algebra, vol. 224, no. 8,pp. 106351, 15, 2020.
🔺The affine funtf variety is the Zariski closure of the set of finite unit norm tight frames. This work characterizes the bases of the algebraic matroid underlying the affine funtf variety of funtfs in R^3, and partial results towards similar characterizations in higher dimensions.🌴[24] Computing Euler obstruction functions using maximum likelihood degrees
(Jose Israel Rodriguez, Botong Wang)
[Arxiv] [DOI].
🔹Int. Math. Res. Not. IMRN, no. 20, pp. 6699–6712, 2020.
🔺Tools from algebraic topology have been used to solve problems in algebraic statistics. In this paper, we go the other way. We use the idea of maximum likelihood degree from algebraic statistics to compute values of the Euler obstruction function in algebraic topology.
Published before August 2020:
🌴[23] A numerical approach for computing Euler characteristics of affine varieties
[Arxiv] [DOI].
🔹Mathematical Software – ICMS 2020(A. M. Bigatti, J. Carette, J. H. Davenport, M. Joswig, and T. de Wolff, eds.), (Cham), pp. 51–60, Springer International Publishing, 2020.
🔺We develop a numerical nonlinear algebra approach for computing the Euler characteristic of an affine variety.🌴[22] Algebraic Optimization Degree
(Härkönen, Marc; Hollering, Benjamin; Kashani, Fatemeh Tarashi; Rodriguez, Jose Israel)
[Preprint ] [DOI] [PDF].
🔹 ACM Commun. Comput. Algebra 54 (2020), no. 2, 44–48.
🔺This article surveys a Macaulay2 implementation for determining the algebraic degree of an optimization problem. (2020)🌴[21] Numerical Homotopy Methods for Multiparameter Eigenvalue Problems
[DOI].
🔹XXI Householder Symposium on Numerical Linear Algebra, extended abstract
🔺🌴[20] Multiregeneration for polynomial system solving
(Colin Crowley, Jose Israel Rodriguez, Jacob Weiker, Jacob Zoromski)
[Arxiv] [DOI].
🔹ACM Commun. Comput. Algebra, vol. 54, no. 2, pp. 39–43, 2020.
🔺We demonstrate our implementation of a continuation method for solving polynomials systems.🌴[19] Euclidean distance degree of the multiview variety.
(L. G. Maxim, JIR, and B. Wang)
[Arxiv] [DOI].
🔹SIAM J. Appl. Algebra Geom., vol. 4, no. 1, pp. 28–48, 2020.
🔺We solve the open problem in computer vision of determining the Euclidean distance degree of the affine multiview variety.🌴[18] Multiprojective witness sets and a trace test.
(J.D. Hauenstein and JIR).
[Arxiv] [DOI].
🔹Adv. Geom., vol. 20, no. 3, pp. 297–318, 2020.
🔺We generalize regeneration, the trace test, and numerical irreducible decomposition to the multiprojective case. Applications include Alt's problem and tensor decomposition.🌴[17] Homogenized funtf varieties and algebraic frame completion [Preprint] [DOI].
🔹 Extended abstract for ISSAC'18 Poster Session.
🔺We study the homogenized funtf variety and the degrees of its coordinate projections.🌴[16] Solving the likelihood equations to compute Euler obstruction functions [Arxiv] [DOI].
🔹 Mathematical Software - ICMS 2018, Proceedings volume 10931 of Lecture Notes in Computer Science. Springer, pages 405--413, 2018.
🔺This paper describes a Macaulay2 package and algorithms to compute Euler obstructions using maximum likelihood degrees that are determined by solving likelihood equations.🌴[15] Trace test. (Anton Leykin,JIR, Frank Sottile). [Arxiv] [DOI].
🔹Arnold Math. J., vol. 4, no. 1,pp. 113–125, 2018.
🔺We give a brief derivation of the trace test to verify completeness of a partial witness set of an irreducible variety in affine or projective space.🌴[14] The Maximum Likelihood Degree of Toric Varieties
(Carlos Améndola, Nathan Bliss, Isaac Burke, Courtney R. Gibbons, Martin Helmer, Serkan Hoşten, Evan D. Nash,JIR, Daniel Smolkin)
[Arxiv] [DOI].
🔹J. Symbolic Comput., vol. 92, pp. 222–242, 2019.
🔺We study the maximum likelihood degree (ML degree) of toric varieties, known as discrete exponential models in statistics.🌴[13] The maximum likelihood degree of mixtures of independence models
(JIR and Botong Wang)
[Arxiv] [DOI].
🔹SIAM J. Appl. Algebra Geometry, 1(1), 484-506. (23 pages)
🔺We use Euler characteristics to prove an outstanding conjecture by Hauenstein, the first author, and Sturmfels by giving a recursion to determine ML degrees for mixtures of independence models. The recursion has been implemented here.🌴[12] Numerical computation of Galois groups
(J.D. Hauenstein, JIR, F. Sottile)
[Arxiv] [DOI].
🔹Found. Comput. Math., vol. 18, no. 4, pp. 867–890, 2018.
🔺We use numerical homotopy continuation to compute Galois groups. Applications include formation shape control, the Alt-Burmester 4-Bar mechanism, and maximum likelihood estimation.🌴[11] A Probabilistic Algorithm for Computing Data-Discriminants of Likelihood Equations
(JIR, Xiaoxian Tang)
[Arxiv] [DOI].
🔹Journal of Symbolic Computation Volume 83, November-December 2017, Pages 342-364
🔺We develop a probabilistic algorithm with three different strategies for computing Data-Discriminants improving our previous version presented in ISSAC2015.🌴[10] The maximum likelihood data singular locus
( E. Horobeţ and JIR)
[Arxiv] [DOI]
🔹Journal of Symbolic Computation Volume 79, Part 1, March-April 2017, Pages 99-107.
🔺We describe the special locus of data for which the likelihood equations have a solution in the model's singular locus.🌴[9] Critical points via monodromy and local methods
(Abraham Martín del Campo, Jose Israel Rodriguez)
[Arxiv] [DOI]
🔹Journal of Symbolic Computation. Volume 79, Part 3, March-April 2017, Pages 559-574
🔺We use the numerical algebraic geometry tool of monodromy and local methods to compute critical points of the likelihood function and Euclidean distance function.🌴[8] Data discriminants of likelihood equations
(JIR, Xiaoxian Tang)
[Arxiv] [DOI]
🔹ISSAC '15 Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation Pages 307-314.
🔺We develop a probabilistic algorithm for computing data discriminants that is experimentally more efficient than the standard elimination algorithm. Based on the computational results, we propose the real root classification conjecture for the 3 by 3 symmetric matrix model.🌴[7] Maximum likelihood for dual varieties
[Arxiv] [DOI].
🔹SNC '14 Proceedings of the 2014 Symposium on Symbolic-Numeric Computation (2014) Pages 43-49.
🔺MLE for statistical models with discrete data is studied from an algebraic statistics viewpoint. A reformulation of the MLE problem in terms of dual varieties and conormal varieties is given.🌴[6] Maximum likelihood geometry in the presence of data zeros
(Elizabeth Gross and JIR)
[Arxiv] [DOI].
🔹ISSAC '14 Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation (2014) Pages 232-239.
🔺We consider discrete algebraic statistical models and study the solutions to the likelihood equations when the data contain zeros. We give new lower bounds for Maximum Likelihood degrees🌴[5] Maximum Likelihood Duality for Determinantal Varieties
(Jan Draisma, JIR)
[Arxiv] [DOI].
🔹International Mathematics Research Notices, Volume 2014, Issue 20, 1 January 2014, Pages 5648-5666.
🔺We prove that the maximum-likelihood degree of the variety of rank-r matrices equals that of the variety of co-rank (r-1)-matrices; and also establish variants for symmetric and skew-symmetric matrices.🌴[4] Maximum Likelihood for Matrices with Rank Constraints
(J. D. Hauenstein, JIR, B. Sturmfels)
[Arxiv] [DOI].
🔹Journal of Algebraic Statistics, Volume 5, Number 1, Pages 18-38, 2014.
🔺We use numerical algebraic geometry to find maximum likelihood degrees of determinantal varieties and show these techniques can be useful for statisticians.🌴[3] Combinatorial Excess Intersection
[Arxiv] [DOI]
🔹Journal of Symbolic Computation Volume 68, Part 2, May-June 2015, Pages 297-307.
🔺We provide formulas and algorithms for computing the excess numbers of certain ideals. The solution for monomial ideals is given by the mixed volumes of certain polytopes. These results enable us to design specific homotopies for numerical algebraic geometry.🌴[2] A novel method for the solution of the forward displacement problem of spherical parallel manipulators
(JIR and M. Ruggiu)
[DOI].
🔹ZAMM Z. Angew. Math. Mech. Volume 93, Issue 1, January 2013, Pages 73-82.
🔺This work uses algebraic geometry to find an analytical (solvable in radicals) solution to a well studied problem in kinematics.🌴[1] Bounding The Degree of Belyi Polynomials.
[DOI].
🔹J. Number Theory, vol. 133,no. 9, pp. 2892–2900, 2013.
🔺This work began as an undergraduate research project advised by Eric Katz. A new combinatorial argument is given to bound the degree of a Belyi polynomial using the valuation of its roots.