The paper proves uniform approximation for minimal surfaces with applications to a Mittag-Leffler theorem.
problem Approximating complete conformal minimal surfaces with finite curvature.
method Uniform approximation theorem with interpolation for minimal surfaces.
result Obtained a Mittag-Leffler type theorem for minimal immersions.
Lecture notes on homology growth and approximation theorems.
problem Understanding the growth of Betti numbers and related approximation theorems.
method Proof of Lück's approximation theorem, discussion of generalizations, and examination of specific cases.
result Discussion of open problems and approximation theorems for mod- p p p Betti numbers. The paper proves approximation and interpolation theorems for maxfaces with singularities.
problem Proving approximation and interpolation theorems for maxfaces with singularities.
method Surveying and applying Enneper--Weierstrass representation formula methods to maxfaces, incorporating singularity criteria.
result Existence of maxfaces with prescribed singularities and maxfaces with dense image singular set.
Extends Tian theorem to Vaisman manifolds for approximations.
problem Approximating Vaisman metrics by immersions/embeddings.
method Study Vaisman metrics on compact manifolds.
result Extend Tian's theorem to Vaisman manifolds.
Paper connects neural network score approximation to reverse diffusion model distribution approximation.
problem Quantifying the relationship between neural network score approximation and the distribution generated by reverse diffusion models.
method Combines Hornik's universal approximation theorem, Girsanov's theorem, and data processing inequality.
result Neural network score approximation guarantees distribution approximation in reverse diffusion models.
The paper extends L 2 L^2 L 2 -Betti numbers to groups with finite subgroups and introduces approximation theorems.
problem Extending L 2 L^2 L 2 -Betti numbers to groups with finite subgroups. method Theory of characters of infinite groups and character induction from finite subgroups.
result Approximation theorems for L 2 L^2 L 2 -multiplicities. The paper proves deep neural networks with analytic activation can approximate any function.
problem Approximating functions with neural networks using analytic activation functions.
method Elementary proofs for real and complex networks, Stone-Weierstrass theorem, Mergelyan's theorem.
result Closure of neural network classes equals space of polynomials for analytic activation.
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
problem Approximation of differentiable maps on infinite-dimensional manifolds
method Weighted universal approximation theorem
result Universal approximation theorem for differentiable maps
Proves theorem for Riemannian manifolds, extending previous work.
problem Proving Quantitative Fatou Theorem on Riemannian manifolds.
method Extending ε-approximation lemma to manifold setting.
result Proves Quantitative Fatou Theorem for Lipschitz domains on Riemannian manifolds.
Generalizes neural network approximation to infinite-dimensional manifolds and derivatives.
problem Approximating differentiable maps on infinite-dimensional manifolds.
method Proves a weighted Nachbin theorem to establish universal approximation for differentiable maps, including derivatives.
result Linear functions of the signature can approximate path space functionals including their derivatives.
Smooth approximations of Lipschitz maps via Ehresmann fibrations and Reeb sphere theorem for functions.
problem Approximating Lipschitz maps and understanding singular points in Riemannian manifolds.
method Using Ehresmann fibrations and Reeb's sphere theorem for Lipschitz functions.
result A Lipschitz map can be approximated by a smooth map via Ehresmann fibrations.
Universal approximation for stochastic processes using Brownian motion.
problem Approximating stochastic processes with linear functionals.
method Establishing L p L^p L p -type universal approximation theorems for rough path spaces. result Linear functionals on the signature of time-extended Brownian motion can approximate any p p p -integrable stochastic process. The abstract discusses families of holomorphic maps to Oka manifolds with approximation theorems.
problem Approximating \(J_b\)-holomorphic maps to Oka manifolds.
method Constructing continuous or smooth families of \(J_b\)-holomorphic maps to Oka manifolds with approximation on compact Runge sets.
result Runge and Mergelyan approximation theorems and Weierstrass interpolation theorem for families of open Riemann surfaces.
Study approximates Riemannian manifolds using polyhedra.
problem Understanding Tullio Regge's approximation theorem.
method Proof of Regge theorem using polyhedra approximation.
result Integral of scalar curvature approximated by polyhedral curvature.
Paper improves CLT and bootstrap approximations for LSA with decreasing step size.
problem Improving normal approximation and bootstrap methods for LSA with decreasing step sizes.
method Refined Berry-Esseen bounds and multiplier bootstrap procedure for LSA.
result Approximation rates up to 1 / n 1/\sqrt{n} 1/ n for LSA rescaled error distribution. Unified method for CNNs to approximate equivariant maps across various groups.
problem Limited universal approximation theorems for CNNs with specific groups and settings.
method Unified approach to derive universal approximation theorems for equivariant maps by CNNs in diverse settings.
result Ability to handle non-linear equivariant maps between infinite-dimensional spaces for non-compact groups.
DQNs can approximate optimal Q-functions with high accuracy on compact sets.
problem Approximating optimal Q-functions in continuous-time Markov Decision Processes.
method Stochastic control, FBSDEs, residual network approximation theorems, large deviation bounds, viscosity solutions.
result DQNs can approximate optimal Q-functions on compact sets with arbitrary accuracy and high probability.
There are proven few analogues of the Theorem of Moser using The Approximation Theorem of Artin.
In this paper we study the problem of approximation of the L 2 L^2 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…
The paper extends a theorem to number fields without infinite places.
problem Finiteness properties of arithmetic approximate lattices.
method Geometric and homological finiteness properties for countable approximate groups.
result The finiteness length is finite and can be computed explicitly.
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…
Paper proves flexibility of specific relations using convex integration.
problem Holonomic approximation theorem in differential topology.
method Proves the holonomic approximation theorem for first order jets using convex integration.
result Relation is open and ample, leading to flexibility of the theorem.
TVS-FNNs can approximate any continuous function on expanded input spaces.
problem Processing a broader range of inputs like sequences and matrices.
method Proving a universal approximation theorem for TVS-FNNs.
result TVS-FNNs can approximate any continuous function on expanded input spaces.
Unified theorem for deep and shallow joint-equivariant machines.
problem Universal approximation of joint-equivariant machines.
method Constructive universal approximation theorem based on ridgelet transform.
result Unified approximation of deep and shallow networks.
Solves low-rank approximation problems in Hilbert spaces.
problem Low-rank approximation in Hilbert spaces.
method Closed-form solutions and error bounds for bounded linear operators.
result Generalization to bounded linear operators from finite dimensions.
The paper defines new spaces for neural networks and proves approximation theorems.
problem Identifying the appropriate function space and norm for neural network models.
method Defining the Barron space and flow-induced function space, proving approximation theorems.
result Optimal approximation theorems hold for functions in the Barron space for two-layer neural networks and the flow-induced space for residual neural networks.
New geometric proof of convex function differentiability and approximation.
problem Second-order differentiability of convex functions and their approximations.
method Elementary geometric approach to prove classical and recent results.
result New proofs of Lusin approximation of convex functions and bodies by C 1 , 1 C^{1,1} C 1 , 1 functions. 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.
problem Improving approximation of fully connected deep neural networks for optimal convergence rates.
method Deriving approximation bounds specifically for a narrower fully connected deep neural network.
result Achieves an optimal rate (up to a logarithmic factor) for fully connected deep neural networks.
The paper analyzes deep ReLU CNNs' approximation properties in 2D space.
problem Establishing L 2 L^2 L 2 approximation properties for deep ReLU CNNs. method Analysis based on decomposition theorem for convolutional kernels, properties of ReLU activation, and connections with one-hidden-layer ReLU NNs.
result Universal approximation theorem for deep ReLU CNNs with classic structure.
DeepONet learns nonlinear operators from data to identify differential equations.
problem Learning nonlinear operators from data to identify differential equations.
method DeepONet architecture with branch and trunk nets.
result DeepONet significantly reduces generalization error compared to fully-connected networks.
Dropout neural networks can approximate any function with high probability.
problem Approximating functions with dropout neural networks.
method Two universal approximation theorems for dropout neural networks in random and deterministic modes.
result Dropout neural networks can approximate any function in probability and in L q L^q L q . Study analyzes a new algorithm for complex optimization problems.
problem Stochastic bilevel optimisation problems in continuous-time models.
method Continuous-time, two-timescale stochastic approximation algorithm.
result Obtained weak convergence rate using central limit theorem.
Develops manifold calculus for simplicial complexes.
problem Approximating functors from simplicial complexes to topological spaces.
method Adapting manifold calculus to simplicial complexes and proving an approximation theorem.
result Functors can be approximated by polynomial functors under certain conditions.
This paper uses hyperbolic geometry to prove Markov's theorem on irrational numbers and quadratic forms.
problem Classifying worst irrational numbers and indefinite binary quadratic forms with farthest values from zero.
method New proof using hyperbolic geometry, translating between hyperbolic geometry and algebra/number theory.
result Demonstrates how simple closed geodesics and ideal triangulations of the modular torus relate to the problems of straight lines and quadratic forms.
The Shannon theorem is extended to locally compact groups.
problem Identifying the Poisson boundary of locally compact groups.
method Random walks and Shannon-McMillan-Breiman theorem.
result Generalized criteria for identifying Poisson boundaries.
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.
problem Extending the Mittag-Leffler theorem to meromorphic curves and minimal surfaces.
method Established a Mittag-Leffler-type theorem for meromorphic curves and minimal immersions, including interpolation and approximation.
result Complete minimal ends in R^5 are generically embedded, and open Riemann surfaces are characterized for minimal surfaces.
Paper proves Allard's theorem in Alexandrov spaces.
problem Proving Allard's theorem in non-collapsed Alexandrov spaces.
method Developed an intrinsic proof for Riemannian manifolds, then extended to Alexandrov spaces using approximation theorem.
result Explicit constants for constants in terms of geometric data.
The paper develops theory for holomorphic null curves in SL2(C).
problem Developing theory for holomorphic null curves in SL2(C).
method Establish Runge, Mergelyan, Mittag-Leffler, and Carleman type theorems for holomorphic null immersions.
result Proves every open Riemann surface admits a proper holomorphic null embedding into SL2(C).
Improved bounds on neural network expressivity.
problem Understanding neural network expressivity and approximation capabilities.
method Improved bounds on the maximal number of linear regions of ReLU-networks.
result New insights into the expressivity of neural networks.
Study improves accuracy of risk measures using advanced algorithms.
problem Computing accurate risk measures for financial losses.
method Nested stochastic approximation and multilevel acceleration.
result Established central limit theorems for estimation errors.
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 …
Functional input neural networks approximate continuous functions on weighted spaces.
problem Approximating continuous functions on infinite-dimensional weighted spaces.
method Additive family mapping, non-linear activation, linear readouts, Stone-Weierstrass theorem.
result Global universal approximation of continuous functions on weighted spaces.
Study mapping class group action on character varieties, proving Kronecker's Theorem.
problem Topological-dynamical action of mapping class group on character varieties.
method Analyzes T n \Bbb T^n T n -character variety and dense orbit conditions. result Provides a dynamical proof of Kronecker's Theorem.
Method approximates first passage times for birth-death processes.
problem Approximating first passage times for birth-death processes.
method General method using birth-death process properties, Keilson's theorem, and Riemann sums.
result Closed-form expressions for first passage times.
Improves Laplace approximation for Bayesian inference on Riemannian manifolds.
problem Inaccurate Gaussian approximations for complex targets and finite-data posteriors.
method Develops alternative variants of the Laplace approximation using a Riemannian metric.
result Exact approximations at the limit of infinite data, improving practical performance.