Related papers: Effective Minkowski Dimension of Deep Nonparametri…

An additive autoencoder for dimension reduction, which is composed of a serially performed bias estimation, linear trend estimation, and nonlinear residual estimation, is proposed and analyzed. Computational experiments confirm that an…

Denoising Diffusion Probabilistic Models (DDPM) are powerful state-of-the-art methods used to generate synthetic data from high-dimensional data distributions and are widely used for image, audio, and video generation as well as many more…

We apply the Minkowski Tensor statistics to two dimensional slices of the three dimensional density field. The Minkowski Tensors are a set of functions that are sensitive to directionally dependent signals in the data, and furthermore can…

We investigate statistical properties of a likelihood approach to nonparametric estimation of a singular distribution using deep generative models. More specifically, a deep generative model is used to model high-dimensional data that are…

Deep Neural Networks (DNNs) are powerful algorithms that have been proven capable of extracting non-Gaussian information from weak lensing (WL) data sets. Understanding which features in the data determine the output of these nested,…

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…

We propose an algorithmic framework, that employs active subspace techniques, for scalable global optimization of functions with low effective dimension (also referred to as low-rank functions). This proposal replaces the original…

Recently, Su and Cook proposed a dimension reduction technique called the inner envelope which can be substantially more efficient than the original envelope or existing dimension reduction techniques for multivariate regression. However,…

Statistical inference from high-dimensional data with low-dimensional structures has recently attracted lots of attention. In machine learning, deep generative modeling approaches implicitly estimate distributions of complex objects by…

Sparse linear regression is a central problem in high-dimensional statistics. We study the correlated random design setting, where the covariates are drawn from a multivariate Gaussian $N(0,\Sigma)$, and we seek an estimator with small…

There is increasing interest in the problem of nonparametric regression with high-dimensional predictors. When the number of predictors $D$ is large, one encounters a daunting problem in attempting to estimate a $D$-dimensional surface…

We perform a deeper analysis of an axiomatic approach to the concept of intrinsic dimension of a dataset proposed by us in the IJCNN'07 paper (arXiv:cs/0703125). The main features of our approach are that a high intrinsic dimension of a…

Supervised manifold learning methods learn data representations by preserving the geometric structure of data while enhancing the separation between data samples from different classes. In this work, we propose a theoretical study of…

The problem of estimating, from a random sample of points, the dimension of a compact subset $S$ of the Euclidean space is considered. The emphasis is put on consistency results in the statistical sense. That is, statements of convergence…

Generalization theory has been established for sparse deep neural networks under high-dimensional regime. Beyond generalization, parameter estimation is also important since it is crucial for variable selection and interpretability of deep…

The geometric structure of an optimization landscape is argued to be fundamentally important to support the success of deep neural network learning. A direct computation of the landscape beyond two layers is hard. Therefore, to capture the…

Unsupervised dimensionality reduction is one of the commonly used techniques in the field of high dimensional data recognition problems. The deep autoencoder network which constrains the weights to be non-negative, can learn a low…

In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes…

This article contributes to the current statistical theory of deep neural networks (DNNs). It was shown that DNNs are able to circumvent the so--called curse of dimensionality in case that suitable restrictions on the structure of the…

We analyze recurrent neural networks with diagonal hidden-to-hidden weight matrices, trained with gradient descent in the supervised learning setting, and prove that gradient descent can achieve optimality \emph{without} massive…