arXiv research

The paper proves approximation and interpolation theorems for maxfaces with singularities.

Extends Tian theorem to Vaisman manifolds for approximations.

The paper proves deep neural networks with analytic activation can approximate any function.

It was pointed out to us that the proof of a crucial lemma (Lemma 5.3) in the paper is incorrect. Thus the approximation theorem (Theorem 0.1) for L^2 torsion of an amenable covering of a finite simplicial complex remains unproved. However, results and proofs of the first four sections (in particular, the approximation…

Universal approximation theorem for differentiable maps on infinite-dimensional manifolds

Proves theorem for Riemannian manifolds, extending previous work.

Generalizes neural network approximation to infinite-dimensional manifolds and derivatives.

Universal approximation for stochastic processes using Brownian motion.

The abstract discusses families of holomorphic maps to Oka manifolds with approximation theorems.

Study approximates Riemannian manifolds using polyhedra.

Paper improves CLT and bootstrap approximations for LSA with decreasing step size.

There are proven few analogues of the Theorem of Moser using The Approximation Theorem of Artin.

Unified method for CNNs to approximate equivariant maps across various groups.

In this paper, we prove a uniform approximation theorem with interpolation for complete conformal minimal surfaces with finite total curvature in the Euclidean space $\mathbb{R}^n$ $(n\ge 3)$. As application, we obtain a Mittag-Leffler type theorem for complete conformal minimal immersions $M\to\mathbb{R}^n$ on any ope…

DQNs can approximate optimal Q-functions with high accuracy on compact sets.

In this paper we study the problem of approximation of the $L^2$-topological invariants by their finite dimensional analogues. We obtain generalizations of the theorem of Lück, dealing with towers of finitely sheeted normal coverings. We prove approximation theorems, establishing relations between the homological invar…

Skeleta and other pure subsets of manifold stratified spaces are shown to have neighborhoods which are teardrops of stratified approximate fibrations (under dimension and compactness assumptions). In general, the stratified approximate fibrations cannot be replaced by bundles, and the teardrops cannot be replaced by ma…

The paper extends a theorem to number fields without infinite places.

Paper proves flexibility of specific relations using convex integration.

TVS-FNNs can approximate any continuous function on expanded input spaces.

These short lecture notes provide a brief introduction to the field of homology growth. They are composed out of two lectures, which I have given at the Borel seminar 2017 in Les Diablerets. We give a proof of Lück's approximation theorem, discuss generalizations and mention some related open problems. Then we discuss …

Unified theorem for deep and shallow joint-equivariant machines.

Let $G$ be a group with a finite subgroup $H$. We define the $L^2$-multiplicity of an irreducible representation of $H$ in the $L^2$-homology of a proper $G$-CW-complex. These invariants generalize the $L^2$-Betti numbers. Our main results are approximation theorems for $L^2$-multiplicities which extend the approximati…

New geometric proof of convex function differentiability and approximation.

We provide a critical analysis of the proof of the fundamental theorem of asset pricing given in the paper "Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing" by B. Wong and C.C. Heyde (Stochastics, 2010) in the context of incomplete Itô-process models. We show that their approach can only w…

This paper improves deep neural network approximation for fully connected networks, achieving optimal convergence rates.

The paper analyzes deep ReLU CNNs' approximation properties in 2D space.

While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator. This universal approximation theorem is suggestive of the pot…

Dropout neural networks can approximate any function with high probability.

Study analyzes a new algorithm for complex optimization problems.

Although stochastic approximation learning methods have been widely used in the machine learning literature for over 50 years, formal theoretical analyses of specific machine learning algorithms are less common because stochastic approximation theorems typically possess assumptions which are difficult to communicate an…

The Mittag-Leffler theorem is extended to meromorphic curves and minimal surfaces.

Paper proves Allard's theorem in Alexandrov spaces.

This work provides closed-form solutions and minimum achievable errors for a large class of low-rank approximation problems in Hilbert spaces. The proposed theorem generalizes to the case of bounded linear operators the previous results obtained in the finite dimensional case for the Frobenius norm. The theorem provide…

The paper develops theory for holomorphic null curves in SL2(C).

Study improves accuracy of risk measures using advanced algorithms.

Improved bounds on neural network expressivity.

We prove a version of the classical Runge and Mergelyan uniform approximation theorems for non-orientable minimal surfaces in Euclidean 3-space R3. Then, we obtain some geometric applications. Among them, we emphasize the following ones: 1. A Gunning-Narasimhan type theorem for non-orientable conformal surfaces. 2. An …

Study mapping class group action on character varieties, proving Kronecker's Theorem.

Functional input neural networks approximate continuous functions on weighted spaces.

In this paper, we prove a Donsker type approximation theorem for the Rosenblatt process, which is a selfsimilar stochastic process exhibiting long range dependence. By using numerical results and simulated data, we show that this approximation performs very well. We use this result to construct a binary market model dr…

Improves Laplace approximation for Bayesian inference on Riemannian manifolds.

Complex-valued neural networks can approximate any continuous function.

We derive results on the distribution of directions of saddle connections on translation surfaces using only the Birkhoff ergodic theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give an alternative proof of a weak version of…

Real analytic maps can be unstable even if infinitesimal changes are stable.

NODEs can approximate a wide range of diffeomorphisms with strong guarantees.

This study presents a method for constructing a sequence of approximate solutions of increasing accuracy to general equilibrium models on nonlocal domains. The method is based on a technique originated from dynamical systems theory. The approximate solutions are constructed employing the Contraction Mapping Theorem and…

This is an introductory text on the more topological aspects of contact geometry, written for the Handbook of Differential Geometry vol. 2. After discussing (and proving) some of the fundamental results of contact topology (neighbourhood theorems, isotopy extension theorems, approximation theorems), I move on to a deta…