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Recently, deep learning has been widely applied in functional data analysis (FDA) with notable empirical success. However, the infinite dimensionality of functional data necessitates an effective dimension reduction approach for functional…

It is often of interest to assess whether a function-valued statistical parameter, such as a density function or a mean regression function, is equal to any function in a class of candidate null parameters. This can be framed as a…

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…

This paper presents a novel approach for the output range estimation problem in Deep Neural Networks (DNNs) by integrating a Simulated Annealing (SA) algorithm tailored to operate within constrained domains and ensure convergence towards…

This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of…

Due to the non-convex nature of training Deep Neural Network (DNN) models, their effectiveness relies on the use of non-convex optimization heuristics. Traditional methods for training DNNs often require costly empirical methods to produce…

We consider the problem of learning a target function corresponding to a deep, extensive-width, non-linear neural network with random Gaussian weights. We consider the asymptotic limit where the number of samples, the input dimension and…

Stochastic gradient descent (SGD) is a popular algorithm for optimization problems arising in high-dimensional inference tasks. Here one produces an estimator of an unknown parameter from independent samples of data by iteratively…

Recent advances in statistical inference have significantly expanded the toolbox of probabilistic modeling. Historically, probabilistic modeling has been constrained to (i) very restricted model classes where exact or approximate…

In this paper, the estimation problem for sparse reduced rank regression (SRRR) model is considered. The SRRR model is widely used for dimension reduction and variable selection with applications in signal processing, econometrics, etc. The…

For various applications, the relations between the dependent and independent variables are highly nonlinear. Consequently, for large scale complex problems, neural networks and regression trees are commonly preferred over linear models…

Deep neural networks (DNNs) trained with the logistic loss (i.e., the cross entropy loss) have made impressive advancements in various binary classification tasks. However, generalization analysis for binary classification with DNNs and…

We develop a minimax rate analysis to describe the reason that deep neural networks (DNNs) perform better than other standard methods. For nonparametric regression problems, it is well known that many standard methods attain the minimax…

The deep neural network suffers from many fundamental issues in machine learning. For example, it often gets trapped into a local minimum in training, and its prediction uncertainty is hard to be assessed. To address these issues, we…

Approximate Bayesian inference on the basis of summary statistics is well-suited to complex problems for which the likelihood is either mathematically or computationally intractable. However the methods that use rejection suffer from the…

Nonlinear differential equations are challenging to solve numerically and are important to understanding the dynamics of many physical systems. Deep neural networks have been applied to help alleviate the computational cost that is…

A main puzzle of deep neural networks (DNNs) revolves around the apparent absence of "overfitting", defined in this paper as follows: the expected error does not get worse when increasing the number of neurons or of iterations of gradient…

Neural networks (NNs) are primarily developed within the frequentist statistical framework. Nevertheless, frequentist NNs lack the capability to provide uncertainties in the predictions, and hence their robustness can not be adequately…

We study deep neural networks and their use in semiparametric inference. We establish novel rates of convergence for deep feedforward neural nets. Our new rates are sufficiently fast (in some cases minimax optimal) to allow us to establish…

Polynomial regression is a recurrent problem with a large number of applications. In computer vision it often appears in motion analysis. Whatever the application, standard methods for regression of polynomial models tend to deliver biased…