Related papers: Any Deep ReLU Network is Shallow

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 investigate the asymptotic properties of deep Residual networks (ResNets) as the number of layers increases. We first show the existence of scaling regimes for trained weights markedly different from those implicitly assumed in the…

Feed-forward networks can be interpreted as mappings with linear decision surfaces at the level of the last layer. We investigate how the tangent space of the network can be exploited to refine the decision in case of ReLU (Rectified Linear…

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…

Survival analysis models time-to-event distributions with censorship. Recently, deep survival models using neural networks have dominated due to their representational power and state-of-the-art performance. However, their "black-box"…

Parameter space is not function space for neural network architectures. This fact, investigated as early as the 1990s under terms such as ``reverse engineering," or ``parameter identifiability", has led to the natural question of parameter…

This paper presents a new mathematical framework to analyze the loss functions of deep neural networks with ReLU functions. Furthermore, as as application of this theory, we prove that the loss functions can reconstruct the inputs of the…

This paper considers the power of deep neural networks (deep nets for short) in realizing data features. Based on refined covering number estimates, we find that, to realize some complex data features, deep nets can improve the performances…

Deep neural networks (DNNs) may outperform human brains in complex tasks, but the lack of transparency in their decision-making processes makes us question whether we could fully trust DNNs with high stakes problems. As DNNs' operations…

Deep convolutional networks provide state of the art classifications and regressions results over many high-dimensional problems. We review their architecture, which scatters data with a cascade of linear filter weights and non-linearities.…

Multiple scales coexist in complex networks. However, the small world property makes them strongly entangled. This turns the elucidation of length scales and symmetries a defiant challenge. Here, we define a geometric renormalization group…

In practice, deeper networks tend to be more powerful than shallow ones, but this has not been understood theoretically. In this paper, we find the analytical solution of a three-layer network with a matrix exponential activation function,…

We point out that (continuous or discontinuous) piecewise linear functions on a convex polytope mesh can be represented by two-hidden-layer ReLU neural networks in a weak sense. In addition, the numbers of neurons of the two hidden layers…

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…

Deluge Networks (DelugeNets) are deep neural networks which efficiently facilitate massive cross-layer information inflows from preceding layers to succeeding layers. The connections between layers in DelugeNets are established through…

The formalism of complex networks is extensively employed to describe the dynamics of interacting agents in several applications. The features of the connections among the nodes in a network are not always provided beforehand, hence the…

LASSO regularization is a popular regression tool to enhance the prediction accuracy of statistical models by performing variable selection through the $\ell_1$ penalty, initially formulated for the linear model and its variants. In this…

This paper considers the growth in the length of one-dimensional trajectories as they are passed through deep ReLU neural networks, which, among other things, is one measure of the expressivity of deep networks. We generalise existing…

Real world data often exhibit low-dimensional geometric structures, and can be viewed as samples near a low-dimensional manifold. This paper studies nonparametric regression of H\"{o}lder functions on low-dimensional manifolds using deep…

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…