We study the problem of maximizing a continuous DR-submodular function that is not necessarily smooth. We prove that the continuous greedy algorithm achieves a [(1-1/e)OPT-ε] guarantee when the function is monotone and Hölder-smooth, meaning that it admits a Hölder-continuous gradient. For functions that are non-differentiable or non-smooth, we propose a variant of the mirror-prox algorithm that attains a [(1/2)OPT-ε] guarantee. We apply our algorithmic frameworks to robust submodular maximization and distributionally robust submodular maximization under Wasserstein ambiguity. In particular, the mirror-prox method applies to robust submodular maximization to obtain a single feasible solution whose value is at least [(1/2)OPT-ε]. For distributionally robust maximization under Wasserstein ambiguity, we deduce and work over a submodular-convex maximin reformulation whose objective function is Hölder-smooth, for which we may apply both the continuous greedy method and the mirror-prox method. This is joint work with Duksang Lee, a fifth-year Ph.D. student at KAIST Math, and Nam Ho-Nguyen from the University of Sydney.

Order types are a combinatorial classification of finite point sets used in discrete and computational geometry. This talk will give an introduction to these objects and their analogue for the projective plane, with an emphasis on their symmetry groups. This is joint work with Emo Welzl: https://arxiv.org/abs/2003.08456

We consider a deep generative model for nonparametric distribution estimation problems. The true data-generating distribution is assumed to possess a certain low-dimensional structure. Under this assumption, we study convergence rates of estimators obtained by likelihood approaches and generative adversarial networks (GAN). The convergence rate depends only on the noise level, intrinsic dimension and smoothness of the underlying structure. The true distribution may or may not possess the Lebesgue density, depending on the underlying structure. For the singular case (no Lebesgue density), the convergence rate of GAN is strictly better than that of the likelihood approaches. Our lower bound of the minimax optimal rates shows that the convergence rate of GAN is close to the optimal rate. If the true distribution allows a smooth Lebesgue density, an estimator obtained by a likelihood approach achieves the minimax optimal rate.

Over recent years, data science and machine learning have been the center of attention in both the scientific community and the general public. Closely tied to the ‘AI-hype’, these fields are enjoying expanding scientific influence as well as a booming job market. In this talk, I will first discuss why mathematical knowledge is important for becoming a good machine learner and/or data scientist, by covering various topics in modern deep learning research. I will then introduce my recent efforts in utilizing various deep learning methods for statistical analysis of mathematical simulations and observational data, including surrogate modeling, parameter estimation, and long-term trend reconstruction. Various scientific application examples will be discussed, including ocean diffusivity estimation, WRF-hydro calibration, AMOC reconstruction, and SIR calibration.