Mateo Díaz (Johns Hopkins): Statistics and DSI Joint Colloquium

Please join us for a Statistics and DSI joint colloquium.

Monday, February 23
11:30am – 12:30pm
Data Science Institute, Room 105
5460 S University Ave, Chicago, IL 60615

Title: Leveraging Structure for Faster Algorithms in Optimization and Diffusion

Abstract: Large-scale iterative methods drive modern AI, yet their theoretical foundations often lag behind their empirical success. We argue that bridging this gap requires identifying the inherent problem structure that enables these algorithms to perform well. This talk instantiates this principle across two domains: optimization and generative modeling.

First, we derive new theoretical guarantees for the Levenberg–Morrison-Marquardt method. Although this method is ubiquitous in settings that demand highly accurate solutions—for instance, when training physics-informed neural networks for scientific discovery—classical guarantees do not explain its strong empirical performance in modern overparameterized, ill-conditioned regimes. By reframing it through the lens of composite optimization, we uncover geometric conditions that ensure fast convergence even in these challenging modern regimes.

Second, we introduce Proximal Diffusion Models (PDM). While standard diffusion models rely on score-matching and forward discretization, we demonstrate that a backward discretization using proximal maps offers significant theoretical and practical advantages. Under mild conditions, we prove that PDM achieves $\varepsilon$-accuracy in KL-divergence within $\widetilde{O}(d/\sqrt{\varepsilon})$ steps and empirically demonstrate that it outperforms conventional methods using fewer sampling iterations.

Bio: I am an Assistant Professor in the Department of Applied Mathematics and Statistics and the Mathematical Institute for Data Science with a secondary appointment in the Department of Computer Science at Johns Hopkins University. Before that, I was a Postdoctoral Scholar at Caltech hosted by Venkat Chandrasekaran and Joel Tropp. I obtained my PhD in Applied Mathematics from Cornell University advised by Damek Davis. I completed a MSc in Mathematics and two BS in Mathematics, and in Systems and Computing Engineering at Universidad de los Andes where I was co-advised by Mauricio Junca and Mauricio Velasco.