相关论文: Approximating Lipschitz continuous functions with …

Low-rank tensor approximations have shown great potential for uncertainty quantification in high dimensions, for example, to build surrogate models that can be used to speed up large-scale inference problems (Eigel et al., Inverse Problems…

We establish universality and expression rate bounds for a class of neural Deep Operator Networks (DON) emulating Lipschitz (or H\"older) continuous maps $\mathcal G:\mathcal X\to\mathcal Y$ between (subsets of) separable Hilbert spaces…

We show expression rates and stability in Sobolev norms of deep feedforward ReLU neural networks (NNs) in terms of the number of parameters defining the NN for continuous, piecewise polynomial functions, on arbitrary, finite partitions…

It is well established that to ensure or certify the robustness of a neural network, its Lipschitz constant plays a prominent role. However, its calculation is NP-hard. In this note, by taking into account activation regions at each layer…

This paper proposes that Lipschitz continuity is a natural outcome of regularized least squares in kernel-based learning. Lipschitz continuity is an important proxy for robustness of input-output operators. It is also instrumental for…

Deep neural networks are notorious for being sensitive to small well-chosen perturbations, and estimating the regularity of such architectures is of utmost importance for safe and robust practical applications. In this paper, we investigate…

We investigate robustness of deep feed-forward neural networks when input data are subject to random uncertainties. More specifically, we consider regularization of the network by its Lipschitz constant and emphasize its role. We highlight…

Neural networks are regularly employed in adaptive control of nonlinear systems and related methods of reinforcement learning. A common architecture uses a neural network with a single hidden layer (i.e. a shallow network), in which the…

Tight estimation of the Lipschitz constant for deep neural networks (DNNs) is useful in many applications ranging from robustness certification of classifiers to stability analysis of closed-loop systems with reinforcement learning…

Graph Neural Networks (GNN) are inherently limited in their expressive power. Recent seminal works (Xu et al., 2019; Morris et al., 2019b) introduced the Weisfeiler-Lehman (WL) hierarchy as a measure of expressive power. Although this…

Rigorous mathematical investigation of learning rates used in back-propagation in shallow neural networks has become a necessity. This is because experimental evidence needs to be endorsed by a theoretical background. Such theory may be…

This paper studies the approximation capacity of neural networks with an arbitrary activation function and with norm constraint on the weights. Upper and lower bounds on the approximation error of these networks are computed for smooth…

We introduce a method to design a computationally efficient $G$-invariant neural network that approximates functions invariant to the action of a given permutation subgroup $G \leq S_n$ of the symmetric group on input data. The key element…

Wasserstein-GANs have been introduced to address the deficiencies of generative adversarial networks (GANs) regarding the problems of vanishing gradients and mode collapse during the training, leading to improved convergence behaviour and…

Recent advances in large-margin classification of data residing in general metric spaces (rather than Hilbert spaces) enable classification under various natural metrics, such as string edit and earthmover distance. A general framework…

In this paper we provide a characterization of intrinsic Lipschitz graphs in the sub-Riemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak…

In recent years, deep learning techniques have shown great success in various tasks related to inverse problems, where a target quantity of interest can only be observed through indirect measurements by a forward operator. Common approaches…

This survey provides an in-depth and explanatory review of the approximation properties of deep neural networks, with a focus on feed-forward and residual architectures. The primary objective is to examine how effectively neural networks…

Providing human-understandable insights into the inner workings of neural networks is an important step toward achieving more explainable and trustworthy AI. Existing approaches to such mechanistic interpretability typically require…

Motivated by classical work on the numerical integration of ordinary differential equations we present a ResNet-styled neural network architecture that encodes non-expansive (1-Lipschitz) operators, as long as the spectral norms of the…