Diffusion models achieve state-of-the-art performance in various generation tasks. However,
their theoretical foundations fall far behind. This paper studies score approximation …

In this article, we review the literature on statistical theories of neural networks from three
perspectives: approximation, training dynamics, and generative models. In the first part …

Learning approximations to smooth target functions of many variables from finite sets of
pointwise samples is an important task in scientific computing and its many applications in …

To improve the accuracy of color image completion with missing entries, we present a
recovery method based on generalized higher-order scalars. We extend the traditional …

We study the problem of training a two-layer neural network (NN) of arbitrary width using
stochastic gradient descent (SGD) where the input $\boldsymbol {x}\in\mathbb {R}^ d $ is …

Neural networks have achieved remarkable empirical performance, while the current
theoretical analysis is not adequate for understanding their success, eg, the Neural Tangent …

Conditional diffusion models serve as the foundation of modern image synthesis and find
extensive application in fields like computational biology and reinforcement learning. In …

Deep nonparametric regression on approximate manifolds: Nonasymptotic error bounds with
polynomial prefactors Page 1 The Annals of Statistics 2023, Vol. 51, No. 2, 691–716 …

In this paper, we study the statistical limits of deep learning techniques for solving elliptic
partial differential equations (PDEs) from random samples using the Deep Ritz Method …

We propose a deep generative approach to sampling from a conditional distribution based
on a unified formulation of conditional distribution and generalized nonparametric …