Related papers: Lipschitz widths
We introduce the concept of average best $m$-term approximation widths with respect to a probability measure on the unit ball of $\ell_p^n$. We estimate these quantities for the embedding $id:\ell_p^n\to\ell_q^n$ with $0<p\le q\le \infty$…
The purpose of this article is to develop a machinery to study the capacity of deep neural networks (DNNs) to approximate high-dimensional functions. In particular, we show that DNNs have the expressive power to overcome the curse of…
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…
In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for…
In this paper, we study the nonhomogeneous Dirichlet problem concerning general semilinear elliptic equations in divergence form. We establish that the boundary Lipschitz regularity of solutions under some more weaker conditions on the…
We give a condition on the curvature tensors of Riemannian manifolds that admit Lipschitz approximation by polyhedral metrics with curvature bounded below or above. We show that this condition is also sufficient for the existence of local…
In this paper, we study approximation properties of single hidden layer neural networks with weights varying on finitely many directions and thresholds from an open interval. We obtain a necessary and at the same time sufficient measure…
Understanding geometric properties of natural language processing models' latent spaces allows the manipulation of these properties for improved performance on downstream tasks. One such property is the amount of data spread in a model's…
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…
We study the structure of the space of coarse Lipschitz maps between Banach spaces. In particular we introduce the notion of norm attaining coarse Lipschitz maps. We extend to the case of norm attaining coarse Lipschitz equivalences, a…
In recent years, there has been a strong interest in measuring the available bandwidth of network paths. Several methods and techniques have been proposed and various measurement tools have been developed and evaluated. However, there have…
We present a general framework, treating Lipschitz domains in Riemannian manifolds, that provides conditions guaranteeing the existence of norming sets and generalized local polynomial reproduction - a powerful tool used in the analysis of…
We revisit the mean field parametrization of shallow neural networks, using signed measures on unbounded parameter spaces and duality pairings that take into account the regularity and growth of activation functions. This setting directly…
For finite-dimensional problems, stochastic approximation methods have long been used to solve stochastic optimization problems. Their application to infinite-dimensional problems is less understood, particularly for nonconvex objectives.…
The recovery of the intrinsic geometric structures of data collections is an important problem in data analysis. Supervised extensions of several manifold learning approaches have been proposed in the recent years. Meanwhile, existing…
In this article, we introduce the Lipschitz bounded approximation property for operator ideals. With this notion, we extend the original work done by Godefroy and Kalton and give some partial answers on the equivalence between the bounded…
Many elliptic boundary value problems exhibit an interior regularity property, which can be exploited to construct local approximation spaces that converge exponentially within function spaces satisfying this property. These spaces can be…
The Lipschitz constant of a neural network is connected to several important properties of the network such as its robustness and generalization. It is thus useful in many settings to estimate the Lipschitz constant of a model. Prior work…
Motivated by the approach of random linear codes, a new distance in the vector space over a finite field is defined as the logarithm of the "surface area" of a Hamming ball with radius being the corresponding Hamming distance. It is named…
Reduced basis methods for approximating the solutions of parameter-dependant partial differential equations (PDEs) are based on learning the structure of the set of solutions - seen as a manifold ${\mathcal S}$ in some functional space -…