相关论文: Constructive Universal High-Dimensional Distributi…

A recurrent neural network (RNN) is a widely used deep-learning network for dealing with sequential data. Imitating a dynamical system, an infinite-width RNN can approximate any open dynamical system in a compact domain. In general, deep…

This paper develops simple feed-forward neural networks that achieve the universal approximation property for all continuous functions with a fixed finite number of neurons. These neural networks are simple because they are designed with a…

This paper considers the problem of regression over distributions, which is becoming increasingly important in machine learning. Existing approaches often ignore the geometry of the probability space or are computationally expensive. To…

This paper proves an abstract theorem addressing in a unified manner two important problems in function approximation: avoiding curse of dimensionality and estimating the degree of approximation for out-of-sample extension in manifold…

We derive universal approximation results for the class of (countably) $m$-rectifiable measures. Specifically, we prove that $m$-rectifiable measures can be approximated as push-forwards of the one-dimensional Lebesgue measure on $[0,1]$…

We investigate the approximation capabilities of dense neural networks. While universal approximation theorems establish that sufficiently large architectures can approximate arbitrary continuous functions if there are no restrictions on…

We introduce a novel generative model, the Discrete Distribution Networks (DDN), that approximates data distribution using hierarchical discrete distributions. We posit that since the features within a network inherently capture…

Score-based generative modeling (SGM) has grown to be a hugely successful method for learning to generate samples from complex data distributions such as that of images and audio. It is based on evolving an SDE that transforms white noise…

We develop a projected Wasserstein distance for the two-sample test, a fundamental problem in statistics and machine learning: given two sets of samples, to determine whether they are from the same distribution. In particular, we aim to…

The goal of this thesis is to improve our understanding of the internal mechanisms by which deep artificial neural networks create meaningful representations and are able to generalize. We focus on the challenge of characterizing the…

Score-based generative models are trained in high-dimensional ambient spaces, yet many data distributions are supported on low-dimensional nonlinear structures. We prove that, for compact $d$-dimensional smooth manifolds $\mathcal{M}…

We address the problem of efficiently computing Wasserstein distances for multiple pairs of distributions drawn from a meta-distribution. To this end, we propose a fast estimation method based on regressing Wasserstein distance on sliced…

Many traditional computer vision algorithms generate realistic images by requiring that each patch in the generated image be similar to a patch in a training image and vice versa. Recently, this classical approach has been replaced by…

We study the expressivity of one-dimensional (1D) ReLU deep neural networks through the lens of their linear regions. For randomly initialized, fully connected 1D ReLU networks (He scaling with nonzero bias) in the infinite-width limit, we…

We develop a kernel projected Wasserstein distance for the two-sample test, an essential building block in statistics and machine learning: given two sets of samples, to determine whether they are from the same distribution. This method…

We introduce a deep learning model that can universally approximate regular conditional distributions (RCDs). The proposed model operates in three phases: first, it linearizes inputs from a given metric space $\mathcal{X}$ to $\mathbb{R}^d$…

Score-based Generative Models (SGMs) approximate a data distribution by perturbing it with Gaussian noise and subsequently denoising it via a learned reverse diffusion process. These models excel at modeling complex data distributions and…

We build a new class of generative algorithms capable of efficiently learning an arbitrary target distribution from possibly scarce, high-dimensional data and subsequently generate new samples. These generative algorithms are particle-based…

Deep neural networks (DNNs) have emerged as a popular mathematical tool for function approximation due to their capability of modelling highly nonlinear functions. Their applications range from image classification and natural language…

Using Stein's method techniques introduced by Chatterjee (2008) and further extended by Kasprzak and Peccati (2022) and by Lachi\`eze-Rey and Peccati (2017), we derive novel quantitative bounds on the convergence in distribution of…