Research

I am working with a random ensemble of recursively defined orthogonal matrices called random butterfly matrices. Their recursive structure has been used to accelerate common methods in computational linear algebra, which has led to a recent spike in interest in butterfly matrices in the machine learning and artificial intelligence communities. Butterfly structures have been integrated into software architecture used in learning fast solvers for large linear systems and in image recognition. I am interested in getting a larger picture of the numerical, spectral and group properties of butterfly matrices. This was the main target for my dissertation of the same name ("Numerical, spectral and group properties of random butterfly matrices"). Currently, I am looking at the impact on growth factors of using butterfly matrices to preprocess linear systems, properties of butterfly Hadamard matrices, applications to randomized sketching, as well as using butterfly matrices to approximate Haar orthogonal matrices. 

In a project with Tom Trogdon, we explored the particular application of using random butterfly matrices to remove the need of (partial) pivoting. Although partial pivoting involes fewer comparisons (O(n²)) than the leading order complexity of GE (i.e., O(n³)), in practice the cost of the actual associated data movements can lead to significant slowdowns in total computation time (~20%). Using butterfly matrices to remove the need of pivoting -- which can be done for "free" since matrix-matrix computations with butterfly matrices take O(n² log n) FLOPs, smaller again than GE complexity -- was first established by D. Stott Parker in 1995, and has been implemented in fast linear solver software as recently as 2023 (e.g., Julia). We studied what might be lost by removing pivoting in terms of stability by empirically comparing growth factors and relative errors using GE with certain pivoting schemes after using Parker's preprocessing strategy. We also further provide a full distributional description of the growth factors for Haar-butterfly matrices, which was a significant improvement in the studies of random growth factors that previously were limited to empirical moment estimations.

A preprint for our paper ``Growth factors of random butterfly matrices and the stability of avoiding pivoting'' is available on arXiv in March 2022. A summary of our results is also available in the poster I shared at the Conference on Random Matrix Theory and Numerical Linear Algebra at the University of Washington in June 2022. This work is now published in the SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 3, 945-970 (2023).

Gaussian elimination (GE) with partial pivoting (GEPP) remains the most common method to solve dense linear systems, which results in a PA=LU factorization of an input matrix A, where P is a permutation matrix, L a lower unipotent triangular matrix (with no entries larger than 1 in magnitude), and U an upper triangular matrix. If A is a random matrix, then P yields a random associated permutation. Each GE step uses a row transposition if needed to ensure the leading pivot entry is maximal in magnitude for the first column of the remaining untriangularized subblock. I wanted to study how often this pivot movement is needed using particular random ensembles. I provide explicit distributions for the number of pivot movements needed using GEPP for some particular random ensembles. A natural output provides representatives for matrices of the form PL for which GEPP uses maximal pivot movements (viz., when P represents an n-cycle). Experiments estimating the empirical spectral density (ESD) of PL lead to a new conjecture on a universality class of random matrices with fixed sparsity whose scaled ESD converges to a measure on the complex unit disk that is an interpolation of the uniform measure on the unit disk and the Dirac measure at the origin.

I am further studying universality properties related to this random ensemble with sparsity and its variants in a project with Xiucai Ding, while I am also studying other statistical properties (e.g., longest increasing subsequence) of random permutations generated using GEPP on certain random matrix ensembles in a project with Chenyang Zhong.

A preprint for "Distribution of the number of pivots needed using Gaussian elimination with partial pivoting on random matrices" is available on arXiv in January 2023. A summary of my results is available in the poster I shared at the Seminar on Stochastic Processes at the University of Arizona in March 2023. This work is now published in The Annals of Applied Probability, Vol. 34, No. 2, 2294-2325 (April 2024).

I am also very interested in the question of studying worst-case growth when using GE with partial pivoting (GEPP) or complete pivoting (GECP). Although exponential growth is possible with GEPP, growth tends to stay much smaller in practice. Support for this behavior was provided in 2022 by Huang and Tikhomirov’s average-case analysis of GEPP, which showed GEPP growth stays at most polynomial with very high probability when using Gaussian matrices. Research on GECP has also seen a lot of recent interest, with improvements to lower bounds on worst-case GECP growth provided by Edelman and Urschel in early 2023 and a slightly sharpened upper bound by Bisain, Edelman, and Urschel in late 2023. I am interested in studying how GEPP and GECP behave on the same linear systems, with a focus on large growth systems and orthogonal matrices (beyond just butterfly matrices). I am able to improve the worst-case GEPP growth asymptotic (upper and lower) bound on orthogonal matrices. Additionally, I studied how much worse can GECP perform relative to GEPP, which then led to new empirical lower bounds in terms of growth, while I also include an empirical study on a family of exponential GEPP growth matrices whose polynomial behavior in small neighborhoods limits to the initial GECP growth factor.

A preprint for my paper ``Growth factors of orthogonal matrices and local behavior of Gaussian elimination with partial and complete pivoting'' is available on arXiv in September 2023. This paper is accepted and will appear in SIAM Journal of Matrix Analysis and Applications.

I am working on a project with Tom Trogdon and Jim Thomson (Applied Physics Lab, University of Washington) that identifies RMT statistics in the normalized spacings between ocean waves using SWIFT buoy data. In particular, we have shown that normalized spacings between wave peaks follow the same distribution as the spacings between eigenvalues for random matrices sampled from the Gaussian Unitary Ensemble (Wigner surmise) as well as the one-dimensional gas interactions via the Coulomb potential. Such statistics have previously been established for spacings between bus arrivals, parallel parked cars, cars stopped at traffice lights, and NYC subway arrivals, so we extend these examples to include earth systems.

In a project with (previous UA undergrads) Jia Cai and Yunke Wan, we are similarly studying the relationship between the normalized spacing distributions of safety rest areas located on interstate routes. We are able to establish the overall normalized spacings between safety rest areas located on major interstates do match Wigner surmise statistics, while a subset of rest areas in the Rocky Mountain region exhibit Poissonian statistics found in non-interacting particle systems. While previous transportation systems that exhibit Wigner surmise statistics involve moving vehicles (e.g., cars, buses, subway trains), we extend this set to include the dynamics in the networks of stationary facilities. Our preprint "Random matrix statistics and safety rest areas on interstates in the United States" is available on arXiv in September 2023. This paper has been accepted and will appear in the Journal of Statistical Mechanics: Theory and Experiment.

In Fall 2023, I have started an honors thesis project with undergrad Anupama Hazarika to similarly study the spacings between earthquakes in Malaysia (earth systems) and also solar flares (extraterrestial systems).

A matrix is totally positive (TP) if all of its minors are positive. TP matrices have been studied extensively for almost a century, and maintain many desirable numerical and spectral properties (e.g., positive eigenvalues, osciallatory eigenvectors). In in project with Nick Ercolani and Jon Ramalheira-Tsu, we are studying a particular ensemble of TP matrices that combines attributes from several other well studied matrix forms, which together form the  Totally positive Hessenberg Toeplitz (TPHT) matrices. This further combines the two well-studied matrix ensembles of Totally positive Hessenberg matrices and Hessenberg Toeplitz matrices. Treated separately, each ensemble brings forward a rich set of spectral tools (such as the Grenader-Szegő theorem from Toeplitz theory) that combined bring forth strong linear algebraic and spectral structures, which we outline and highlight new results and observations. Through experiments we further study the spectral and numerical properties of these matrices, while ongoing work will further expand  this lens to include the k-Chop and Lusztig factors. Additionally, we will study the properties of new random ensembles from this subclass. 

Our preprint "Total positivity and spectral theory for Toeplitz Hessenberg matrix ensembles" is available on arXiv in November 2023.

Starting in 2019, I have provided analysis for the UCI Math CEO program. I built a reporting structure to efficiently determine the impact of Math CEO participation on student proficiency metrics, which indicated increases in both math and reading comprehension among participants. I will lead a longitudinal analysis project using using this structure to determine the effect of student reading and math proficiency on continued participation in Math CEO for a fixed cohort, as well as a survey analysis project looking at the impact of COVID-19 and the resultant shift to an online learning on the program and its participants.

In December 2020, we were awarded the Inclusive Excellence Spirit Award to support this last project. In April 2021, I was awarded a Summer Inclusive Excellence Grant by the UCI Grad Division to support my continued work with Math CEO. This work is joint with Alessandra Pantano.

In 2022-23, I am working on a project with Melinda Lanius (Auburn) on using universal course design concepts in building, designing and maintaining an online, asynchronous linear algebra course. We submitted our paper, "Through Vexatious Visualizations and Mounding Modalities: Chasing Universal Design in a Linear Algebra Course", in February 2023.

I am interested in applying tools and results in RMT to problems in graph theory and number theory. In particular, I am interested in studying the rank of the Laplacian for bipartite random graphs reduced over a finite field. This will yield information about the generators of the associated sandpile group. This is a joint project with Nathan Kaplan.

While at the Social and Economic Sciences Research Center, I was involved in a number of research projects with district and state departments that looked into post-secondary college readiness factors, cyber-bullying, and living wage analysis. These projects resulted in one educational journal publication ("Northshore graduates' high school performance and need for remediation in college", available in the Standard Deviation), as well as a large number of technical reports.

I also am interested in studying the mathematical properties in ranked-choice voting models. I gave presentations in seminars for undergraduates at the University of Arizona and Pepperdine University that included an overview of the instant run-off model and the Tideman method (and the Condorcet winner), with an application of both methods using ranked-choice submissions by seminar participants to determine: Who is the best Spider-Man? (Here is a link to the presentation as well as the R code used during the talk to determine the winners using both models.)