UCSB Applied Math/PDE/Data Science Seminar

Abstract: In this talk, we introduce LOT Wassmap, a computationally feasible algorithm to uncover low-dimensional structures in the Wasserstein space. The algorithm is motivated by the observation that many datasets are naturally interpreted as probability measures rather than points in $\mathbb{R}^n$, and that finding low-dimensional descriptions of such datasets requires manifold learning algorithms in the Wasserstein space. Most available algorithms are based on computing the pairwise Wasserstein distance matrix, which can be computationally challenging for large datasets in high dimensions. Our algorithm leverages approximation schemes such as Sinkhorn distances and linearized optimal transport to speed-up computations, and in particular, avoids computing a pairwise distance matrix. Experiments demonstrate that LOT Wassmap attains correct embeddings and that the quality improves with increased sample size. We also show how LOT Wassmap significantly reduces the computational cost when compared to algorithms that depend on pairwise distance computations.

Host: Katy Craig

Abstract: K-means clustering a widely used machine learning method to group data points based on their similarity. Solving the K-means problem is computationally hard and various tractable (or even scalable) approximation algorithms (Lloyd, spectral, nonnegative matrix factorization, semidefinite programming) have been explored in theory and practice. In this talk, I will discuss some recent developments from my research group on the statistical optimality and computational guarantee for relaxed formulations of K-means. To break the computational bottleneck, we leverage convex and non-convex optimization techniques to tackle the clustering problem and present their achievement of an information-theoretic sharp threshold for exact recovery of cluster labels under the standard Gaussian mixture model. Joint work with Yun Yang (University of Maryland), Yubo Zhuang (UIUC & USC), Richard Y. Zhang (UIUC).

Host: Sui Tang

Abstract: Deep learning has made significant advancements over the past decade. This talk aims to explore deep learning within the framework of reproducing kernel Banach spaces (RKBSs). We begin by considering learning problems in a general functional context, establishing explicit and data-dependent representer theorems for both minimal norm interpolation (MNI) and regularization problems. These theorems serve as a foundational basis for subsequent results in deep learning. We then introduce a hypothesis space that employs deep neural networks (DNNs), treating the DNN as a function of two variables: the physical variable and the parameter variable. This leads to the construction of a vector-valued RKBS, whose reproducing kernel is the product of the DNN and a weight function. Within this framework, we analyze MNI and regularization problems, derive corresponding representer theorems, and present solutions in the form of DNNs.

Host: Sui Tang

Abstract: Most existing theoretical investigations of the accuracy of diffusion models, albeit significant, assume the score function has been approximated to a certain accuracy, and then use this a priori bound to control the error of generation. This article instead provides a first quantitative understanding of the whole generation process, i.e., both training and sampling. More precisely, it conducts a non-asymptotic convergence analysis of denoising score matching under gradient descent. In addition, a refined sampling error analysis for variance exploding models is also provided. The combination of these two results yields a full error analysis, which elucidates (again, but this time theoretically) how to design the training and sampling processes for effective generation. For instance, our theory implies a preference toward noise distribution and loss weighting in training that qualitatively agree with the ones used in [Karras et al. 2022]. It also provides perspectives on the choices of time and variance schedules in sampling: when the score is well trained, the design in [Song et al. 2020] is more preferable, but when it is less trained, the design in [Karras et al. 2022] becomes more preferable.

Host: Bohan Zhou

Abstract: This talk will describe certain families of metrics between functions defined over a variety of geometries. The metrics we will consider are provably robust to deformations of their inputs, and can be rapidly evaluated. While they are related to transportation distances between probability measures, these metrics are more flexibly defined between functions with negative values and unequal integrals. Theoretical properties will be illustrated with selected numerical experiments.

Host: Katy Craig

Abstract: Metalenses are ultra thin surfaces that are composed of nano structures to focus light. These nano structures manipulate light waves by abrupt phase shifts over the scale of the wavelength to bend them in unusual ways. Compared to the bulky, thick shapes of the conventional lenses, metalenses offer many advantages in optical applications due to their reduced thicknesses and multifunctionalities. Mathematically a metalens can be represented by a pair (Γ, Φ) where Γ is a surface in R^3, and Φ is a C^1 function defined in a neighborhood of Γ, called phase discontinuity. The knowledge of Φ yields the type of arrangements of the nano structures on the surface that are needed for a specific refraction job. In this talk we are going to discuss some refraction problems starting from the existence of phase discontinuity functions that refract a ray in desired directions. Later on, using methods from optimal transport, we will discuss the existence of energy conserving phases for any given general directions.

Host: Elie Abdo

Abstract: In this talk we discuss partial data inverse boundary problems for magnetic Schrödinger operators on bounded domains in the Euclidean space as well as some Riemannian manifolds with boundary. In particular, we show that the knowledge of the set of the Cauchy data on a portion of the boundary of a domain in the Euclidean space of dimension $n\ge 3$ for the magnetic Schrödinger operator with a magnetic potential of class $W^{1,n}\cap L^\infty$, and an electric potential of class $L^n$, determines uniquely the magnetic field as well as the electric potential.  Our result is an extension of global uniqueness results of Dos Santos Ferreira--Kenig--Sjöstrand--Uhlmann (2007) and Knudsen--Salo (2007), to the case of less regular electromagnetic potentials. Our approach is based on boundary Carleman estimates for the magnetic Schrödinger operator, regularization arguments, as well as the invertibility of the geodesic X-ray transform.  In this talk, we will also show an extension of our uniqueness result to non-admissible manifolds. This talk is based on joint-work with Lili Yan.

Host: Hanming Zhou

Abstract: The hydrostatic Euler equations, also known as the inviscid primitive equations, are utilized to describe the motion of inviscid fluid flow in a thin domain, such as the ocean and atmosphere on a planetary scale. It is well-known that this model is ill-posed in Sobolev spaces and Gevrey class of order greater than one, and its analytic solutions can form singularity in finite time. In this talk, I will present two recent results on the stochastic version of this model. Firstly, with general multiplicative noise, I will demonstrate how the local Rayleigh condition can be used to address the issue of ill-posedness in Sobolev spaces, leading to the establishment of the existence and uniqueness of pathwise solutions. Secondly, I will discuss that some specific random noises can restore the local well-posedness and prevent finite-time blowups. These are joint works with Ruimeng Hu and Rongchang Liu.

Host: Elie Abdo