Research
Research
Here are some examples of questions that are interesting to me:
If you fit a function or a manifold to some data points, what kind of geometric properties does this function or manifold have?
If a portion of your data is corrupted, to what extent can you recover it as faithfully as possible? What if you have "help"?
Below, you can find a set of evolving tools I have used in my work, roughly organized into three boxes.
My dissertation research is on Whitney-type extension and selection problems and their connection to machine learning. At the basic level, Whitney's problems ask for efficient and universal ways to fit a function to random data while minimizing energy. Here is an expository article by Charles Fefferman with an overview of Whitney's problems. I am particularly interested in how nonlinear constraints, such as positivity or convexity, preserve or alter the structure of these problems.
A problem posed by H. Feichtinger (and subsequently by C. Heil and D. Larson) asks whether a positive-definite integral operator with M1 (Feichtinger algebra) kernel admits a rank-one decomposition series that is also strongly square-summable in M1. Intuitively, one can think of it as a version of Mercer's Theorem, but in L1. One can formulate the finite-dimensional variant of the problem in terms of optimal matrix factorization in l1. In a recent preprint with R. Balan, complemented by a concurrent result by Bandeira-Mixon-Steinerberger, we showed that the answer is no in general, and we studied special cases when a desired factorization is possible.
The Gromov-Wasserstein (GW) distance is a mathematical framework for comparing two metric measure spaces by aligning their intrinsic geometric structures, making it suitable for analyzing datasets with different underlying domains. In machine learning, GW distance is used in tasks like domain adaptation, graph matching, and shape analysis. Still, it has been observed that the GW distance is inherently sensitive to outlier noise and cannot accommodate partial matching. In a recent preprint with T. Needham et al., we study the approximate metric properties when one relaxes the marginal matching and the stability of these approximations against empirical contamination.
(with T. Needham et al.) Metric properties of partial and robust Gromov-Wasserstein distances, 2025. arXiv:2411.02198
(with R. Balan) Factorization of positive-semidefinite operators with absolutely summable entries, 2024. arXiv:2409.20372
Roots, trace, and extendability of flat nonnegative smooth functions, International Mathematics Research Notices, online, 2024. arXiv:2306.16183
(with C. Fefferman and G.K. Luli) C^2 interpolation with range restriction, Revista Matemática Iberoamericana, 39(2):649--710, 2023. arXiv:2107.08272
(with C. Liang, Y. Liang, and G.K. Luli) Univariate Range-Restricted C^2 Interpolation Algorithms, Journal of Computational and Applied Mathematics, 425:115040, 2023.
(with G.K. Luli and K. O'Neill) Smooth selection for infinite sets, Advances in Mathematics, 407:108566, 2022. arXiv:2109.04905
(with G.K. Luli and K. O'Neill) On the shape fields finiteness principle, International Mathematics Research Notices, online, 2021. arXiv:2010.09827
(with G.K. Luli) Algorithms for nonnegative C^2(R^2) interpolation, Advances in Mathematics, 385:107756, 2021. arXiv:2102.05777
(with G.K. Luli) C^2(R^2) nonnegative extension by bounded-depth operators, Advances in Mathematics, 375:107391, 2020. arXiv:2008.04962
(with G.K. Luli) Nonnegative C^2(R^2) interpolation, Advances in Mathematics, 375:107364, 2020. arXiv:1901.09876
Nonnegative Whitney Extension Problem for C^1(R^n), 2019. arXiv:1912.06327
(with K. Xu et al.) A Transfer Function Approach to Shock Duration Compensation for Laboratory Evaluation of Ultra-High-G Vacuum-Packaged MEMS Accelerometers, IEEE 32nd International Conference on Micro Electro Mechanical Systems (MEMS), 676--679, 2019.
(with K. Xu et al.) Micromachined integrated self-adaptive nonlinear stops for mechanical shock protection of MEMS, Journal of Micromechanics and Microengineering, 28:064006, 2018.
(with K. Xu et al.) Micromachined integrated shock protection via a self-adaptive nonlinear system, 19th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS), 524--527, 2017.
Joint Mathematics Meeting (JMM), Special Session on the Mathematics of Adversarial, Interpretable, and Explainable AI, January 8-11, 2025, Seattle, WA.
One World MINDS Seminar, November 21, 2024, online.
Frame Theory Day, Saint Louis University, October 18-20, St. Louis, MI.
AMS Postdoc Seminar, Johns Hopkins University, October 11, Baltimore, MD
Loo-Keng Hua Lecture, Chinese Academy of Sciences, July 6-7, 2024, Beijing, China.
Mathematics of Adversarial, Interpretable, and Explainable AI, Mathematics Research Communities (MRC) conference, June 23-29, 2024. Buffalo, NY.
AMS Spring Southern Sectional Meeting, Special Session on Bases and Frames in Hilbert Spaces III, March 24, 2024, Florida State University.
Research Interaction Team (RIT) on Applied Harmonic Analysis, February 19-26, 2024, University of Maryland.
The 15th Whitney Workshop, August 14-25, 2023. UC Davis (Online).
Conference for early-career mathematical researchers, June 18-20, 2023, Shenzhen University.
CUNY GC Harmonic Analysis and PDE Seminar, March 10, 2023. CUNY Graduate Center.
ICERM semester seminar, October 14, 2022. ICERM at Brown University.
Fall Fourier Talks 2022, Oct 6-7, 2022. Norbert Wiener Center at the University of Maryland (poster).
Harmonic analysis methods in geometric tomography, Sep. 26-30, 2022. ICERM at Brown University.
14th Whitney Workshop, May 20 - November 11, 2021. Thursdays 7:30 AM PST. UC Davis (Online).
Bay Graduate Math Conference, May 11, 2019, University of California, Berkeley.