Research
I am a mathematical research scientist interested in data fitting and predictive modeling. In the broadest terms, my research focuses on making optimal decisions from limited information. Here are some examples of questions that are interesting to me:
- How can we efficiently approximate a function or a manifold from a finite set of data points?
- How can we compare two datasets that are defined on different spaces?
- How can we recover a corrupted dataset as faithfully as possible?
Whitney Extension Theory
At the basic level, Whitney Problems ask for efficient methods to fit a smooth function to random data points while minimizing some norms or energies and obeying certain constraints (e.g. nonnegativity, convexity). Here are two sample flavors:
- Problem 1: Given a finite set of points \(E \subset \mathbb{R}^n\) and a function \(f : E \to \mathbb{R}\), how can we construct a nonnegative smooth function that interpolates these points while minimizing the \(C^m\) or Sobolev \(L^m_p\) norm?
- Problem 2: How can we fit a low-dimensional manifold \(\Sigma\) to a pointcloud \(E\subset \mathbb{R}^n\) with optimal geometric properties, i.e., \(\mathrm{curv}(\Sigma)\) as small as possible?
Optimal Transport
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 \begin{equation} \mathcal{D}_{\epsilon,p}(X,Y)^p:= \inf_{\pi \in \mathcal{P}(X\times Y), \pi_X \sim_{\epsilon} \mu_X, \pi_Y \sim_{\epsilon} \mu_Y} \mathbb{E}_{\pi\otimes \pi}[|d_X - d_Y|^p] \end{equation} and the stability of these approximations against empirical contamination.
Frames in Banach Spaces
A problem posed by H. Feichtinger (and subsequently by C. Heil and D. Larson) asks whether a positive-definite integral operator with \(\mathcal{M}_{1}\) (Feichtinger algebra) kernel admits a rank-one decomposition series that is also strongly square-summable in \(\mathcal{M}_{1}\), i.e., \begin{equation} T = \sum_{k=1}^{\infty} f_k^*\otimes f_k \quad \text{with} \quad \sum_{k=1}^\infty \|f_k\|_{\mathcal{M}_1}^2 < \infty \end{equation} Intuitively, one can think of it as a version of Mercer's Theorem, but in \(L_1\). One can formulate the finite-dimensional variant of the problem in terms of optimal matrix factorization in \(\ell_1^n\). 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.