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Neural networks are a powerful class of functions that can be trained with simple gradient descent to achieve state-of-the-art performance on a variety of applications. Despite their practical success, there is a paucity of results that…
In analyzing of modern biological data, we are often dealing with ill-posed problems and missing data, mostly due to high dimensionality and multicollinearity of the dataset. In this paper, we have proposed a system based on matrix…
Is a sample rich enough to determine, at least locally, the parameters of a neural network? To answer this question, we introduce a new local parameterization of a given deep ReLU neural network by fixing the values of some of its weights.…
We discuss the expressive power of neural networks which use the non-smooth ReLU activation function $\varrho(x) = \max\{0,x\}$ by analyzing the approximation theoretic properties of such networks. The existing results mainly fall into two…
A main theoretical interest in biology and physics is to identify the nonlinear dynamical system (DS) that generated observed time series. Recurrent Neural Networks (RNNs) are, in principle, powerful enough to approximate any underlying DS,…
Activation functions are non-linearities in neural networks that allow them to learn complex mapping between inputs and outputs. Typical choices for activation functions are ReLU, Tanh, Sigmoid etc., where the choice generally depends on…
Typical methods for supervised sequence modeling are built upon the recurrent neural networks to capture temporal dependencies. One potential limitation of these methods is that they only model explicitly information interactions between…
For neural networks (NNs) with rectified linear unit (ReLU) or binary activation functions, we show that their training can be accomplished in a reduced parameter space. Specifically, the weights in each neuron can be trained on the unit…
Fully connected deep neural networks are successfully applied to classification and function approximation problems. By minimizing the cost function, i.e., finding the proper weights and biases, models can be built for accurate predictions.…
Neural networks with REctified Linear Unit (ReLU) activation functions (a.k.a. ReLU networks) have achieved great empirical success in various domains. Nonetheless, existing results for learning ReLU networks either pose assumptions on the…
We study the expressivity of deep neural networks. Measuring a network's complexity by its number of connections or by its number of neurons, we consider the class of functions for which the error of best approximation with networks of a…
In this study, we establish that deep neural networks employing ReLU and ReLU$^2$ activation functions can effectively represent Lagrange finite element functions of any order on various simplicial meshes in arbitrary dimensions. We…
Estimation of a regression function from independent and identically distributed data is considered. The $L_2$ error with integration with respect to the distribution of the predictor variable is used as the error criterion. The rate of…
Deep ReLU Networks can be decomposed into a collection of linear models, each defined in a region of a partition of the input space. This paper provides three results extending this theory. First, we extend this linear decompositions to…
Active learning methods aim to improve sample complexity in machine learning. In this work, we investigate an active learning scheme via a novel gradient-free cutting-plane training method for ReLU networks of arbitrary depth and develop a…
The training process of neural networks usually optimize weights and bias parameters of linear transformations, while nonlinear activation functions are pre-specified and fixed. This work develops a systematic approach to constructing…
We prove a large deviation principle for deep neural networks with Gaussian weights and at most linearly growing activation functions, such as ReLU. This generalises earlier work, in which bounded and continuous activation functions were…
Rectified Linear Units (ReLU) seem to have displaced traditional 'smooth' nonlinearities as activation-function-du-jour in many - but not all - deep neural network (DNN) applications. However, nobody seems to know why. In this article, we…
This paper focuses on establishing $L^2$ approximation properties for deep ReLU convolutional neural networks (CNNs) in two-dimensional space. The analysis is based on a decomposition theorem for convolutional kernels with a large spatial…
Despite the empirical success of DNN, their internal training dynamics remain difficult to characterize. In ReLU-based models, the activation pattern induced by a given input determines the piecewise-linear region in which the network…