Mathematics of Data Science Seminar meets (usually) on Tuesdays 2:00-3:00. The schedule is below. If you would like to speak in the seminar, please contact Maria Han Veiga (hanveiga.1@osu.edu) or Vladimir Kobzar (kobzar.1@osu.edu).
| Date | Room | Speaker | Title |
| Fall 2025 | |||
| September 16, 2:00 | Daniela Calvetti (Case Western Reserve University) | Dictionary learning: where inverse problems and data science meet | |
| September 30, 2:00 | JR0221 | Emily King (Colorado State) | Some Group Actions in Data Science |
| October 7, 2:00 | CL0109 | Wei Zhu (Georgia Tech) | Structure-preserving machine learning and data-driven structure discovery |
| October 28, 2:00 | Eric Goubault (LIX, Ecole Polytechnique) | Verification of Cyber-Physical Systems | |
| Spring 2026 | |||
| March 10, 2:00 | Timur Akhunov (Wabash College) | Hypoelliptic PDE-based characterization of Thompson sampling | |
| March 27, 2:00 | Jakob Zech (University of Heidelberg) | Statistical and Bayesian Learning of Infinite-Dimensional PDE Operators | |
| March 30, 2:00 | Lianghao Cao (Caltech) | Learning History-Dependent Constitutive Models | |
| April 21, 2:00 | João Pereira (University of Georgia) | Multi-subspace power method for decomposing all tensors |
Title: Dictionary learning: where inverse problems and data science meet
Abstract: In the current era of big data, a lot of effort is spent in organizing and querying data sets. These efforts are particularly central for inverse problems, in which the goal is to estimate quantities depending indirectly on the data. Previously collected or simulated data sets provide an insight into how typical data should look like, and how much variability in the data can be expected as the unknown of interest varies. In this context, it is natural to think of data sets as entries in a dictionary. Intrinsic knowledge about the data combined with data science methods can be used to partition the dictionary into subdictionaries. Matching previously unseen data to labeled dictionary entries can provide an interpretation of the data. Dictionary matching/learning methods provide a flexible and versatile framework for traditional classification problems as well as for solving inverse problems where traditional techniques fail either because the forward model is complex, ill-defined, or difficult to parametrize, or the data are insufficient for standard methods. To increase computational efficiency and accuracy dictionary matching can be preceded by a dictionary learning step, yielding a reduced dictionary. Sparsity of the solutions can greatly speed up the calculation and to facilitate the interpretation of the dictionary matching process. Hierarchical Bayesian methods leading to very effective computations with sparsity-promoting prior models are very naturally suited for dictionary learning applications. In this talk, we review the sparsity promoting methods for solving inverse problems as a dictionary matching problem, and discuss how Bayesian modeling error methods and matrix factorization techniques can be used to learn compressed dictionaries.
Title: Some Group Actions in Data Science
Abstract: Although neural networks yield powerful results in many applications, there are drawbacks to their use. For example, there are many fields where the amount of labeled data needed to train neural networks is insufficient (i.e., in the realm of un- and semi-supervised learning). Further, the internal decision-making process of a neural network is often opaque, leading to a desire for more interpretable techniques. These concerns have led some to consider mathematics like group actions, in both algorithms built on harmonic analysis and invariant and equivariant learning. Some preliminary theoretical results concerning "partial" invariance and equivariance will also be presented.
Title: Structure-preserving machine learning and data-driven structure discovery
Abstract: Many machine learning and scientific computing tasks, including computer vision and the computational modeling of physical and engineering systems, have intrinsic structures. Empirical studies demonstrate that models incorporating these structures often achieve significantly improved performance. Meanwhile, there is growing interest in discovering structures directly from observational data. In this talk, I will present our recent works on the interplay between structure and data. I will discuss how specific structures can be efficiently embedded into machine learning models and rigorously quantify the resulting performance gains. Furthermore, I will explore techniques for discovering structures, such as conservation laws, integrability, and Lax pairs, from observational physical data.
Title: Verification of Cyber-Physical Systems
Abstract: In this talk, we are going to present some of the set-based methods we have been designing over the last few years for formally verifying important aspects of cyber-physical systems. Cyber-physical systems are controlled systems, defined mathematically by a collection of ODEs in general, and of discrete systems (programs or even neural nets) that define the controllers involved. Concentrating on non-distributed aspects, we will show how to compute inner and outer approximations of the set of reachable states for these systems. The outer-approximations allow at least to verify some basic properties such as converging to the target state with sufficient precision under some given time, and proving that the system never enters a dangerous configuration. The inner-approximation allow also to disprove such properties, in particular.
In the first part of the talk, we will describe the way we use zonotopes, generalized mean-value theorems and Taylor methods in order to properly inner and outer approximate the set of reachable states for a CPS, possibly controlled by a neural network with differentiable activation function. Some benchmarks will be discussed as well, using our RINO tool.
More complex properties, such as reach-avoid properties and robust reachability under perturbations need not only inner and outer approximations of reachable states, but also approximations of much more complicated sets, that are definable in quantified first-order logic. We will show how to generalize the methods that we described in the first part of the talk, to these more general « generalized reachability » problems, first using simple interval methods, and then, more generally, using methods of arbitrary order. This has also be implemented in a prototype, and some benchmarks will be discussed as well.
In the final part of the talk, we will show that these methods can also be extended so as to deal with non-deterministic as well as probabilistic uncertainties. This relies on the theory of imprecise probabilities and we will show first results in that direction, including on classical benchmarks in verification of neural networks, such as the ACAS Xu, where we show that our approach improves the tradeoff between tightness and efficiency compared to state of the art related work on probabilistic network verification. If time permits, we will show some preliminary results about using topology (more precisely, TDA) instead of probabilities, to assert robustness of neural networks.
Title: Hypoelliptic PDE-based characterization of Thompson sampling
Abstract: The multi-armed bandit is a foundational problem in reinforcement learning with extensive industrial applications. Named after a gambler seeking the most rewarding slot machine, it epitomizes the exploration-exploitation tradeoff. Thompson sampling is a classic strategy to navigate this tradeoff and minimize the number of selections of suboptimal arm (regret) over the long run. In classical applications, expected rewards of the arms differ by a fixed, if unknown, amount. In recent problems motivated by the large samples used in machine learning applications, however, such rewards diminish over time, leaving few foundational results known. While the bandit games are inherently discrete, in joint work with Vlad Kobzar, we have characterized the continuous-time limit of this problem by a degenerate parabolic PDE. We verified that this PDE satisfies Hörmander's bracket condition and is hypoelliptic. Is hypoellipticity enough to guarantee regret bounds for our gambler? We will share known results and open questions from our work in progress.
Title: Statistical and Bayesian Learning of Infinite-Dimensional PDE Operators
Abstract: Learning infinite-dimensional operators, such as the solution maps of parametric PDEs, from finite noisy data is a central challenge in scientific machine learning. A naive application of classical statistical learning theory in this setting suffers from the curse of dimensionality. In this talk, we present a theoretical framework for operator learning. We establish well-posedness and sample complexity rates for Empirical Risk Minimization (ERM) and the Bayesian posterior mean under an infinite-dimensional white noise model. Our formulation allows us to exploit the holomorphic regularity of the underlying PDE operators to bypass the curse of dimensionality, yielding dimension-independent convergence rates.
Title: Learning History-Dependent Constitutive Models
Abstract: History-dependent constitutive models are central to material modeling because macroscopic material response often depends on loading history through memory effects emerging from microscale physics. Since these macroscopic constitutive laws are rarely available in closed form and their parameters are not directly measurable, they must be learned from data. This talk presents two complementary approaches to this learning problem, using data from microscopic and macroscopic scales.
First, we consider operator learning of homogenized constitutive laws from microscopic unit-cell simulations, capturing dependence on both deformation history and microstructural properties. Here, the focus is on characterizing the expressivity and complexity of a neural-operator architecture that combines the internal-variable formalism and Fourier neural operators.
Second, we consider learning constitutive parameters from macroscopic material testing, where parameter identifiability is often a major challenge. We show that a Bayesian optimal experimental design framework can be used to discover optimal designs of specimen geometries and loading paths that substantially reduce uncertainty in the identified parameters relative to randomly selected designs, especially for parameters associated with memory effects.
Title: Multi-subspace power method for decomposing all tensors
Abstract: I will discuss a recent method we proposed for calculating the canonical polyadic decomposition of tensors. The method applies to low rank tensors with any symmetry pattern, from fully symmetric to asymmetric tensors, recovering tensor components that respect the tensor symmetries. We reduce the problem of tensor decomposition to finding tensor singular vectors, which we calculate using a variation of the shifted higher order power method (SS-HOPM), which we propose and establish its global convergence. Numerical experiments demonstrate that our decomposition algorithm is robust to noise and achieves higher accuracy and faster runtime than existing methods.
| Date | Speaker | Title |
| Fall 2024 | ||
| September 10, 11:30 | Bernardo Modenesi (University of Michigan) | Unveiling Hidden Patterns in Agent Behavior with Discrete-Choice and Network Theory |
| October 17, 16:00 (Thursday) | Tim Kunisky (John Hopkins University) | Spectral pseudorandomness, free probability, and the clique number of the Paley graph |
| November 19, 11:30 | Thomas O'Leary-Roseberry (UT Austin) | Efficient Infinite-Dimensional Bayesian Inversion using Derivative-Informed Neural Operators |
| Spring 2025 | ||
| March 25, 11:30 | Nguyen-Truc-Dao Nguyen (San Diego State University) | Optimization of Model Predictive Control using iLQR and Neural Network |
| April 17, 11:30 | Paul Laiu (Oak Ridge National Lab) | Accelerated and communication efficient federated learning algorithms |
| April 22, 11:30 | Vladimir Kobzar (Ohio State University) | Fourier-Based Bounds for Wasserstein Distances and Their Applications in Data-Driven Problems |
Title: Unveiling Hidden Patterns in Agent Behavior with Discrete-Choice and Network Theory
Abstract: Many datasets in data science stem from agents making repeated choices over time, with each choice leading to an observable outcome. In this setup, we introduce a novel approach to uncover latent agent heterogeneity, enhancing our understanding of agent behavior and improving causal inference estimation. By combining discrete choice models with network theory, we develop a method to measure agent similarity based on their patterns of choice. This results in a network-based unsupervised clustering technique that groups agents with similar behaviors, offering an interpretable alternative to black-box machine learning clustering models, with explicit estimation assumptions. In this seminar, I will illustrate our approach using labor market data, where workers (agents) and jobs (choices) are represented as nodes in a bipartite network, with edges in this network denoting worker-job matches. By clustering workers based on their job choices, we can infer unobserved workers' skills—an important factor in economic analysis. Through Bayesian estimation, we reveal latent groups of similar workers, which is used to make more accurate predictions of labor market outcomes and measurement of labor market discrimination, compared to models relying only on observable characteristics. This seminar will detail our methodological framework, estimation strategy, and practical applications for understanding and predicting agent-choice dynamics.
Title: Spectral pseudorandomness, free probability, and the clique number of the Paley graph
Abstract: The Paley graph is a classical number-theoretic construction of a graph that is believed to behave "pseudorandomly" in many regards. Accurately bounding the clique number of the Paley graph is a long-standing open problem in number theory, with applications to several other questions about the statistics of finite fields. I will present a new approach to this problem, which also opens up intriguing connections with random matrix theory and free probability. In particular, I will show that certain deterministic induced subgraphs of the Paley graph have the same limiting spectrum as induced subgraphs on random subsets of vertices of the same size. I will discuss how this phenomenon arises as a consequence of asymptotic freeness (in the sense of free probability) of certain matrices associated with the Paley graph. I will then present conjectures describing a stronger analogy between random and pseudorandom deterministic induced subgraphs that would lead to clique number bounds improving on the state of the art. On the way, I will describe new techniques for understanding the eigenvalue statistics of more general random or pseudorandom submatrices of certain structured matrices like ones associated to incoherent tight frames, and will mention how these helped to resolve a recent conjecture of Haikin, Zamir, and Gavish in frame theory.
Title: Efficient Infinite-Dimensional Bayesian Inversion using Derivative-Informed Neural Operators
Abstract: We address Bayesian inverse problems (BIPs) for functions that parametrize PDE models, such as heterogeneous coefficient fields. These problems are challenging due to high-dimensional parameter spaces upon discretization, the computational demands of PDE-based likelihood evaluations, and the complex geometry of the posterior (e.g., concentration and non-linear multi-modality). While efficient algorithms for infinite-dimensional BIPs leverage posterior geometry through likelihood derivatives, they become impractical when the computational model (and its adjoints) is expensive to compute. In this talk, we introduce derivative-informed neural operators (DINOs), a class of neural operator surrogates trained to accurately approximate both PDE mappings and their derivatives (Fréchet derivatives). DINOs offer superior generalization over traditional neural operators, enabling precise derivative use in high-dimensional uncertainty quantification (UQ) tasks like BIPs. Specifically, reduced basis DINOs are a natural