Research
Below are some of the research topics that I am currently interested in.
Whitney Extension Theory
At the basic level, Whitney Problems ask for efficient methods to fit a smooth function to unstructured data points (i.e., no asumptions on the distribution or geometry of the data) 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?
\(L_p\) Localization
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 continue to study interesting cases when a desired factorization is possible. The infinite-dimensional problem is intimately related to nuclear and 2-summing operators, and more generally to the geometry of Banach spaces.
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.