相关论文: Minimum Width of Deep Narrow Networks for Universa…

We study the following two related problems. The first is to determine to what error an arbitrary zonoid in $\mathbb{R}^{d+1}$ can be approximated in the Hausdorff distance by a sum of $n$ line segments. The second is to determine optimal…

This paper studies the role of activation functions in learning modular addition with two-layer neural networks. We first establish a sharp expressivity gap: sine MLPs admit width-$2$ exact realizations for any fixed length $m$ and, with…

The developments of deep neural networks (DNN) in recent years have ushered a brand new era of artificial intelligence. DNNs are proved to be excellent in solving very complex problems, e.g., visual recognition and text understanding, to…

In this article we study fully-connected feedforward deep ReLU ANNs with an arbitrarily large number of hidden layers and we prove convergence of the risk of the GD optimization method with random initializations in the training of such…

The expressivity of neural networks as a function of their depth, width and type of activation units has been an important question in deep learning theory. Recently, depth separation results for ReLU networks were obtained via a new…

Deep learning training training algorithms are a huge success in recent years in many fields including speech, text,image video etc. Deeper and deeper layers are proposed with huge success with resnet structures having around 152 layers.…

Deep neural networks with rectified linear units (ReLU) are getting more and more popular due to their universal representation power and successful applications. Some theoretical progress regarding the approximation power of deep ReLU…

Does over-parameterization eliminate sub-optimal local minima for neural networks? An affirmative answer was given by a classical result in [59] for 1-hidden-layer wide neural networks. A few recent works have extended the setting to…

Understanding the representational power of Deep Neural Networks (DNNs) and how their structural properties (e.g., depth, width, type of activation unit) affect the functions they can compute, has been an important yet challenging question…

Overwhelming theoretical and empirical evidence shows that mildly overparametrized neural networks -- those with more connections than the size of the training data -- are often able to memorize the training data with $100\%$ accuracy. This…

We show that there is a simple (approximately radial) function on $\reals^d$, expressible by a small 3-layer feedforward neural networks, which cannot be approximated by any 2-layer network, to more than a certain constant accuracy, unless…

We study two models of ReLU neural networks: monotone networks (ReLU$^+$) and input convex neural networks (ICNN). Our focus is on expressivity, mostly in terms of depth, and we prove the following lower bounds. For the maximum function…

Deep neural networks (DNNs), particularly those using Rectified Linear Unit (ReLU) activation functions, have achieved remarkable success across diverse machine learning tasks, including image recognition, audio processing, and language…

We study finite sample expressivity, i.e., memorization power of ReLU networks. Recent results require $N$ hidden nodes to memorize/interpolate arbitrary $N$ data points. In contrast, by exploiting depth, we show that 3-layer ReLU networks…

Understanding the computational complexity of training simple neural networks with rectified linear units (ReLUs) has recently been a subject of intensive research. Closing gaps and complementing results from the literature, we present…

Whereas recovery of the manifold from data is a well-studied topic, approximation rates for functions defined on manifolds are less known. In this work, we study a regression problem with inputs on a $d^*$-dimensional manifold that is…

Recent Progress has shown that exploitation of hidden layer neurons in convolution neural networks incorporating with a carefully designed activation function can yield better classification results in the field of computer vision. The…

Training deep neural networks is a challenging non-convex optimization problem. Recent work has proven that the strong duality holds (which means zero duality gap) for regularized finite-width two-layer ReLU networks and consequently…

We prove sharp dimension-free representation results for neural networks with $D$ ReLU layers under square loss for a class of functions $\mathcal{G}_D$ defined in the paper. These results capture the precise benefits of depth in the…

The ubiquity of deep learning algorithms in various applications has amplified the need for assuring their robustness against small input perturbations such as those occurring in adversarial attacks. Existing complete verification…