Related papers: Quantile regression with deep ReLU Networks: Estim…
This article concerns the expressive power of depth in neural nets with ReLU activations and bounded width. We are particularly interested in the following questions: what is the minimal width $w_{\text{min}}(d)$ so that ReLU nets of width…
Deep learning searches for nonlinear factors for predicting asset returns. Predictability is achieved via multiple layers of composite factors as opposed to additive ones. Viewed in this way, asset pricing studies can be revisited using…
This paper establishes the (nearly) optimal approximation error characterization of deep rectified linear unit (ReLU) networks for smooth functions in terms of both width and depth simultaneously. To that end, we first prove that…
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 $[0,1]$…
In this paper we are concerned with the approximation of functions by single hidden layer neural networks with ReLU activation functions on the unit circle. In particular, we are interested in the case when the number of data-points exceeds…
We consider a family of deep neural networks consisting of two groups of convolutional layers, a downsampling operator, and a fully connected layer. The network structure depends on two structural parameters which determine the numbers of…
In this paper, we consider estimation and inference for the unknown parameters and function involved in a class of generalized hierarchical models. Such models are of great interest in the literature of neural networks (such as Bauer and…
Learning to predict solutions to real-valued combinatorial graph problems promises efficient approximations. As demonstrated based on the NP-hard edge clique cover number, recurrent neural networks (RNNs) are particularly suited for this…
Motivated by challenges in conditional generative modeling, where the target conditional density takes the form of a ratio f1 over f2, this paper develops a theoretical framework for approximating such ratio-type functionals. Here, f1 and…
We investigate the training and generalization errors of overparameterized neural networks (NNs) with a wide class of leaky rectified linear unit (ReLU) functions. More specifically, we carefully upper bound both the convergence rate of the…
In this paper, we investigate the relationship between deep neural networks (DNN) with rectified linear unit (ReLU) function as the activation function and continuous piecewise linear (CPWL) functions, especially CPWL functions from the…
Deep learning empirically achieves high performance in many applications, but its training dynamics has not been fully understood theoretically. In this paper, we explore theoretical analysis on training two-layer ReLU neural networks in a…
This note presents a simple way to add a count (or quantile) constraint to a regression neural net, such that given $n$ samples in the training set it guarantees that the prediction of $m<n$ samples will be larger than the actual value (the…
Each year, deep learning demonstrates new and improved empirical results with deeper and wider neural networks. Meanwhile, with existing theoretical frameworks, it is difficult to analyze networks deeper than two layers without resorting to…
Deep Neural Networks (DNNs) typically require massive amount of computation resource in inference tasks for computer vision applications. Quantization can significantly reduce DNN computation and storage by decreasing the bitwidth of…
While deep learning is successful in a number of applications, it is not yet well understood theoretically. A satisfactory theoretical characterization of deep learning however, is beginning to emerge. It covers the following questions: 1)…
We study the approximation properties of shallow neural networks with an activation function which is a power of the rectified linear unit. Specifically, we consider the dependence of the approximation rate on the dimension and the…
We consider the problem of finding a two-layer neural network with sigmoid, rectified linear unit (ReLU), or binary step activation functions that "fits" a training data set as accurately as possible as quantified by the training error; and…
We consider the embedding of piecewise-linear deep neural networks (ReLU networks) as surrogate models in mixed-integer linear programming (MILP) problems. A MILP formulation of ReLU networks has recently been applied by many authors to…
ReLU neural-networks have been in the focus of many recent theoretical works, trying to explain their empirical success. Nonetheless, there is still a gap between current theoretical results and empirical observations, even in the case of…