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The expressive power of neural networks is important for understanding deep learning. Most existing works consider this problem from the view of the depth of a network. In this paper, we study how width affects the expressiveness of neural…

In a previous work we have detailed the requirements to obtain a maximal performance benefit by implementing fully connected deep neural networks (DNN) in form of arrays of resistive devices for deep learning. This concept of Resistive…

Initializing the weights and the biases is a key part of the training process of a neural network. Unlike the subsequent optimization phase, however, the initialization phase has gained only limited attention in the literature. In this…

Activation functions are essential to introduce nonlinearity into neural networks, with the Rectified Linear Unit (ReLU) often favored for its simplicity and effectiveness. Motivated by the structural similarity between a shallow…

This paper investigates efficient deep neural networks (DNNs) to replace dense unstructured weight matrices with structured ones that possess desired properties. The challenge arises because the optimal weight matrix structure in popular…

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 commonly believed that the hidden layers of deep neural networks (DNNs) attempt to extract informative features for learning tasks. In this paper, we formalize this intuition by showing that the features extracted by DNN coincide with…

Recently proposed Gated Linear Networks present a tractable nonlinear network architecture, and exhibit interesting capabilities such as learning with local error signals and reduced forgetting in sequential learning. In this work, we…

Two networks are equivalent if they produce the same output for any given input. In this paper, we study the possibility of transforming a deep neural network to another network with a different number of units or layers, which can be…

Recurrent neural networks (RNNs) are powerful dynamical models, widely used in machine learning (ML) and neuroscience. Prior theoretical work has focused on RNNs with additive interactions. However, gating - i.e. multiplicative -…

We find that different Deep Neural Networks (DNNs) trained with the same dataset share a common principal subspace in latent spaces, no matter in which architectures (e.g., Convolutional Neural Networks (CNNs), Multi-Layer Preceptors (MLPs)…

As deep learning systems are widely adopted in safety- and security-critical applications, such as autonomous vehicles, banking systems, etc., malicious faults and attacks become a tremendous concern, which potentially could lead to…

Deep neural network with rectified linear units (ReLU) is getting more and more popular recently. However, the derivatives of the function represented by a ReLU network are not continuous, which limit the usage of ReLU network to situations…

Activation functions are fundamental to deep neural networks, governing gradient flow, optimization stability, and representational capacity. Within historic deep architectures, while ReLU has been the dominant choice for the activation…

Deep Neural Networks (DNNs) are rapidly being applied to safety-critical domains such as drone and airplane control, motivating techniques for verifying the safety of their behavior. Unfortunately, DNN verification is NP-hard, with current…

While deep neural networks (DNNs) and Gaussian Processes (GPs) are both popularly utilized to solve problems in reinforcement learning, both approaches feature undesirable drawbacks for challenging problems. DNNs learn complex nonlinear…

The brain processes information through many layers of neurons. This deep architecture is representationally powerful, but it complicates learning by making it hard to identify the responsible neurons when a mistake is made. In machine…

Over-parameterized neural networks generalize well in practice without any explicit regularization. Although it has not been proven yet, empirical evidence suggests that implicit regularization plays a crucial role in deep learning and…

Neural networks are complex functions of both their inputs and parameters. Much prior work in deep learning theory analyzes the distribution of network outputs at a fixed a set of inputs (e.g. a training dataset) over random initializations…

The training dynamics and generalization properties of neural networks (NN) can be precisely characterized in function space via the neural tangent kernel (NTK). Structural changes to the NTK during training reflect feature learning and…