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Overparametrization is a key factor in the absence of convexity to explain global convergence of gradient descent (GD) for neural networks. Beside the well studied lazy regime, infinite width (mean field) analysis has been developed for…

Residual connections significantly boost the performance of deep neural networks. However, there are few theoretical results that address the influence of residuals on the hypothesis complexity and the generalization ability of deep neural…

Deep neural networks (DNNs) have significantly advanced machine learning, with model depth playing a central role in their successes. The dynamical system modeling approach has recently emerged as a powerful framework, offering new…

We propose a scalable framework for the learning of high-dimensional parametric maps via adaptively constructed residual network (ResNet) maps between reduced bases of the inputs and outputs. When just few training data are available, it is…

The empirical emergence of neural collapse -- a surprising symmetry in the feature representations of the training data in the penultimate layer of deep neural networks -- has spurred a line of theoretical research aimed at its…

We show the universality of depth-2 group convolutional neural networks (GCNNs) in a unified and constructive manner based on the ridgelet theory. Despite widespread use in applications, the approximation property of (G)CNNs has not been…

Various powerful deep neural network architectures have made great contribution to the exciting successes of deep learning in the past two decades. Among them, deep Residual Networks (ResNets) are of particular importance because they…

Deep learning architectures suffer from depth-related performance degradation, limiting the effective depth of neural networks. Approaches like ResNet are able to mitigate this, but they do not completely eliminate the problem. We introduce…

Modern neural networks are highly overparameterized, with capacity to substantially overfit to training data. Nevertheless, these networks often generalize well in practice. It has also been observed that trained networks can often be…

Despite the widespread empirical success of ResNet, the generalization properties of deep ResNet are rarely explored beyond the lazy training regime. In this work, we investigate \emph{scaled} ResNet in the limit of infinitely deep and wide…

Residual Neural Networks (ResNets) achieve state-of-the-art performance in many computer vision problems. Compared to plain networks without residual connections (PlnNets), ResNets train faster, generalize better, and suffer less from the…

Despite several algorithmic advances in the training of convolutional neural networks (CNNs) over the years, their generalization capabilities are still subpar across several pertinent domains, particularly within open-set tasks often found…

The analogy between Gaussian processes (GPs) and deep artificial neural networks (ANNs) has received a lot of interest, and has shown promise to unbox the blackbox of deep ANNs. Existing theoretical works put strict assumptions on the ANN…

The residual neural network (ResNet) is a popular deep network architecture which has the ability to obtain high-accuracy results on several image processing problems. In order to analyze the behavior and structure of ResNet, recent work…

Learning-based methods have demonstrated remarkable performance in solving inverse problems, particularly in image reconstruction tasks. Despite their success, these approaches often lack theoretical guarantees, which are crucial in…

Standard convolutions are prevalent in image processing and deep learning, but their fixed kernels limits adaptability. Several deformation strategies of the reference kernel grid have been proposed. Yet, they lack a unified theoretical…

Finding parameters in a deep neural network (NN) that fit training data is a nonconvex optimization problem, but a basic first-order optimization method (gradient descent) finds a global optimizer with perfect fit (zero-loss) in many…

Recurrent Neural Networks (RNNs) have been widely applied to sequential data analysis. Due to their complicated modeling structures, however, the theory behind is still largely missing. To connect theory and practice, we study the…

Several image processing tasks, such as image classification and object detection, have been significantly improved using Convolutional Neural Networks (CNN). Like ResNet and EfficientNet, many architectures have achieved outstanding…

We derive finite width and depth corrections for the Neural Tangent Kernel (NTK) of ResNets and DenseNets. Our analysis reveals that finite size residual architectures are initialized much closer to the "kernel regime" than their vanilla…