When a diffusion model is not memorizing the training samples, what does it generate, and why? In this talk, I will describe a quantitative framework for understanding the distribution produced by a learned diffusion model through a data-driven geometric object: a log-density ridge manifold of the smoothed training distribution. This manifold acts as a backbone for generation and reveals a three-stage inference behavior: trajectories first reach the ridge, then align in normal directions, and finally slide along tangent directions. 

This perspective allows us to quantify how training error influences generation in different directions, and to explain when inter-mode generations arise. I will also present a random feature example in which the model’s inductive bias can be decomposed explicitly into architectural bias and optimization error, and tracked along the inference dynamics. Experiments on synthetic multimodal distributions and MNIST latent diffusion support the theory in both low- and high-dimensional settings.

This talk discusses joint work with Venkat Chandrasekaran, Jose Israel Rodriguez, and Kevin Shu, where we initiate the study of Lagrangian dual sections. This theory gives rise to sufficient conditions for the "hidden convexity" of certain nonconvex optimization problems. Notable examples include spectral inverse problems and certain unbalanced Procrustes problems. As an additional bonus, when the constraint set is a compact Riemannian manifold, the Lagrangian formulation allows us to solve these problems using a numerical continuation algorithm based on Riemannian gradient descent.

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Many state-of-the-art methods in machine learning are black boxes which do not allow humans to understand how decisions are made. In a number of applications, like medicine and atmospheric science, researchers do not trust such black boxes. Explainable AI can be thought of as attempts to open the black box of neural networks, while interpretable AI focuses on creating clear boxes. Adversarial attacks are small perturbations of data that cause a neural network to misclassify the data or act in other undesirable ways. Such attacks are potentially very dangerous when applied to technology like self-driving cars. The goal of this talk is to introduce mathematicians to problems they can attack using their favorite mathematical tools. The mathematical structure of transformers, the powerhouse behind large language models like ChatGPT, will also be explained.

Abstract: Generative AI has rapidly transformed machine learning, with diffusion and autoregressive models achieving unprecedented performance across vision, language, and scientific discovery. Despite this success, our theoretical understanding still lags far behind practice: why do these models generalize so effectively from finite data in high dimensions? In this talk, I present a mathematical framework that shows that intrinsic low-dimensional structure is the key to understanding this phenomenon and provides a foundation for building more trustworthy generative AI. Through the lens of mixtures of low-rank Gaussian models, I show that learning high-dimensional distributions can be reduced to a canonical subspace clustering problem. This connection yields provable guarantees: the sample complexity scales with the intrinsic dimension of the data, rather than the ambient dimension, thereby breaking the curse of dimensionality for generalization. I will then turn to the role of representation learning in generalization, using two-layer denoising autoencoders as a tractable model to show that the optimal representations and weight structures differ fundamentally between the memorization and generalization regimes. These results offer a unified perspective on how generative models both learn meaningful structure in latent spaces and synthesize new data in high dimensions. We translate these theoretical insights into practical guidelines for controlled generation, ensuring model safety and privacy. Finally, we conclude by contrasting the generalization performance of diffusion and autoregressive models in the context of state prediction for stochastic dynamical systems. These findings inform new data assimilation methods and provide critical insights across many scientific applications, and establish a foundation for next-generation generative modeling.

Speaker Bio: Qing Qu is an Assistant Professor in EECS at the University of Michigan. He works at the intersection of the foundations of machine learning, numerical optimization, and signal/image processing, with a current focus on the theory of deep generative models and representation learning. Prior to joining Michigan in 2021, he was a Moore–Sloan Data Science Fellow at the Center for Data Science, New York University (2018–2020). He received his Ph.D. in Electrical Engineering from Columbia University in October 2018 and his B.Eng. in Electrical and Computer Engineering from Tsinghua University in July 2011. His work has been recognized with multiple honors, including the Best Student Paper Award at SPARS 2015, a Microsoft PhD Fellowship in Machine Learning (2016), the Best Paper Award at the NeurIPS Diffusion Models Workshop (2023), NSF CAREER Award (2022), Amazon Research Award (AWS AI, 2023), UM CHS Junior Faculty Award (2025), Google Research Scholar Award (2025), and the 1938E Award in Michigan Engineering (2026). He has led and delivered multiple tutorials at ICASSP, CPAL, CVPR, ICCV, and ICML. He was one of the founding organizers and Program Chair for the new Conference on Parsimony & Learning (CPAL), regularly serves as an Area Chair for NeurIPS, ICML, and ICLR, senior area chair for ICASSP’26, and is an Action Editor for TMLR.

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Abstract: We consider stochastic differential equations that are modified by reward functions or likelihood based weights in order to promote specific events. This perspective applies both to diffusion type models used in generative modeling and to SDEs describing physical phenomena such as molecular dynamics or weather. The main emphasis is on rewards that are non smooth or singular, as they appear in conditioning, threshold objectives, and rare event simulation. We discuss diffusion bridges as a central example, where one seeks typical trajectories connecting prescribed endpoints, for instance during a molecular transition between stable states or between two atmospheric configurations. We also discuss fine tuning of diffusion models with non differentiable rewards, motivated by applications that prioritize tail events and other low probability regions.

Bio: Jakiw Pidstrigach is an AI Research Scientist at Gridmatic. He earned his PhD from the University of Potsdam, with research on filtering and diffusion models. He subsequently held a postdoctoral position at the University of Oxford, where he worked on theory and optimal control of AI.

We discuss aspects of generalized matrix inverses from a "consistency-aware" perspective. We show that many standard tools in engineering and applied mathematics (e.g., the SVD) are commonly mis-applied in ways that undermine solution integrity. We then describe straightforward generalizations of these tools that remedy this situation.