Zoom link: https://upenn.zoom.us/j/98220304722

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.