Neural networks can approximate functions uniformly across various measures.
problem Universal approximation of functions across different probability measures.
method Proving neural networks are dense in Orlicz spaces, extending classical theorems.
result Neural networks uniformly approximate functions for weakly compact families of measures.
Deep neural networks can approximate rough functions with high accuracy.
problem Approximating rough functions with neural networks.
method Proved that ENO interpolation can be cast as a deep ReLU neural network, transferring ENO's high-order accuracy.
result Deep neural networks can achieve high-order accuracy in approximating Lipschitz functions.
This work improves sample efficiency in neural function approximation for reinforcement learning.
problem Improving sample efficiency in reinforcement learning with neural function approximation.
method Study of function approximation with two-layer neural networks (ReLU and polynomial activations) under generative and realizability models.
result Significant improvement in sample complexity compared to linear methods.
Deep neural networks with piecewise-polynomial activations can approximate smooth functions and their derivatives.
problem Approximating smooth functions and their derivatives with neural networks.
method Derives the depth, width, and sparsity required for approximation in Hölder norms.
result Deep neural networks with bounded weights can approximate Hölder smooth functions and their derivatives.
Improved neural network approximates analytic and L^p functions efficiently.
problem Efficiently approximating analytic and L^p functions using neural networks.
method Three-dimensional ReLU network architecture for sawtooth functions, improving approximation rates.
result Substantially improved exponential approximation rates for analytic functions and general L^p functions.
Greedy method adds neurons one by one for better function approximation.
problem Function approximation in neural networks.
method Growing deep neural network by adding one neuron at a time with non-linear activation.
result Accurate approximants for model problems in function approximation.
The paper shows neural networks can approximate functions over non-compact domains with non-polynomial activation.
problem Approximating functions over non-compact domains using neural networks.
method Using single-hidden-layer feedforward neural networks with non-polynomial activation functions over non-compact subsets of Euclidean spaces.
result Neural networks can approximate functions in weighted C k C^k C k -spaces and weighted Sobolev spaces over unbounded domains. Paper shows deep neural networks can approximate Korobov functions nearly optimally.
problem Approximating Korobov functions with deep neural networks.
method Used deep neural networks and measured approximation rates with L p L_p L p and H 1 H^1 H 1 norms. result Achieved a super-convergence rate, outperforming traditional methods.
There has been a growing interest in expressivity of deep neural networks. However, most of the existing work about this topic focuses only on the specific activation function such as ReLU or sigmoid. In this paper, we investigate the approximation ability of deep neural networks with a broad class of activation functi…
Quantum neural networks approximate periodic functions more efficiently.
problem Approximating periodic functions with quantum neural networks.
method Using Jackson's inequality to construct a QNN that approximates a trigonometric polynomial of the function.
result Quantum neural networks can achieve better approximation results with fewer parameters for smoother functions.
Quantum neural networks can approximate noisy functions accurately.
problem Approximating noisy functions with quantum neural networks.
method Universal approximation theorem with error bounds for noisy quantum neural networks.
result Quantum neural networks can approximate noisy functions with precise error bounds.
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.
Rational neural networks approximate functions more efficiently with less depth.
problem Choosing optimal nonlinear activation functions in neural networks.
method Rational activation functions with optimal bounds and efficiency proofs.
result Rational neural networks approximate smooth functions more efficiently than ReLU networks with exponentially smaller depth.
Study assesses neural nets for optimization problems, highlighting SiLU's effectiveness.
problem Using neural nets for optimization problems, especially for accurate approximations.
method Determined best activation function (SiLU) for nonlinear optimization problems. Analyzed function approximations using neural networks and interpolation/regression models.
result Neural nets can deliver competitive zero- and first-order approximations but underperform on second-order approximations.
The paper proves neural networks with ReLU and softmax can approximate any function.
problem Approximating functions and class labels in neural networks.
method Extended universal approximator theory to neural networks with ReLU and softmax.
result Neural networks with ReLU and softmax can approximate any function and class labels.
Researchers develop neural networks for approximating functions in Banach spaces.
problem Approximating Banach space valued continuous functions.
method Quasi-interpolation Banach space valued neural network operators using algebraic sigmoid functions.
result Jackson type inequalities for function approximation.
Neural networks can approximate any L^p functions on R^n.
problem Approximating functions on unbounded domains with neural networks.
method Monotone sigmoid, ReLU, ELU, Softplus, LeakyReLU activation functions.
result Shallow neural networks can arbitrarily well approximate L^p functions on R^n.
Survey examines deep neural networks' ability to approximate functions.
problem Approximation of target functions by deep neural networks.
method Examination of feed-forward and residual architectures, focusing on optimization problems in regression and classification.
result Deep neural networks can approximate functions effectively, especially with ReLU activation functions.
Study shows how to approximate and estimate high-dimensional classification functions without the curse of dimensionality.
problem Approximating and estimating classification functions in high-dimensional spaces.
method Modified existing results to show that R B V 2 RBV^2 R B V 2 functions can be approximated by neural networks with bounded weights. Proved the existence of a neural network with bounded weights approximating a classification function. Leveraged these bounds to quantify estimation rates. result Neural networks can approximate R B V 2 RBV^2 R B V 2 functions without the curse of dimensionality, leading to efficient estimation rates. Neural networks can approximate functionals on RKHS with error bounds.
problem Approximating functionals on RKHS using neural networks.
method Interpolating orthogonal projections in RKHS using point evaluations.
result Explicit error bounds for various kernels (inverse multiquadric, Gaussian, Sobolev).
Develops wavelet-based neural network approximation theory.
problem Analyzing neural network approximation capabilities over various activation functions.
method Wavelet frame theory on spaces of homogeneous type, sufficient conditions for approximation, error estimates.
result Derives sufficient conditions for neural networks to approximate any functions in a given space, including non-smooth activations.
We developed safe and ranged approximations for ReLU, tanh, and sigmoid functions to reduce neural network training time.
problem Expensive computation of hyperbolic tangent and sigmoid functions in neural networks.
method Function approximation techniques to create safe and ranged approximations.
result 10% to 37% improvement in training times on CPU and 20% to 53% improvement in ranged cases.
Deep networks can learn functions approximated by shallow networks, but not all functions.
problem The learnability of functions by deep neural networks and the approximation capacity of simpler classes.
method Study the connection between learnability and approximation capacity of functions by deep neural networks and simpler classes.
result A necessary condition for a function to be learnable by deep neural networks is to be approximable by shallow networks.
Paper proves neural networks can be approximated using interval bounds.
problem Verifying safety and robustness of neural networks.
method Introduces interval universal approximation (IUA) theorem for neural networks.
result Neural networks can be approximated using interval bounds for any continuous function and squashable activation functions.
Neural networks with integer weights approximate continuous functions efficiently.
problem Approximating continuous functions using neural networks with integer weights.
method Integrates superexpressive activation functions and integer weights.
result Convergence rate of order n − 2 β 2 β + d log 2 n n^{\frac{-2β}{2β+d}}\log_2n n 2 β + d − 2 β log 2 n for neural network regression. Efficiently approximates neural network function space distance.
problem Estimating the average discrepancy between neural network outputs.
method Linearized Activation Function TRick (LAFTR) for ReLU networks.
result Parametric approximation outperforms nonparametric methods in memory and accuracy.
New neural networks learn distribution functions using quantiles and moments.
problem Approximating functions of distributions in probability spaces.
method Quantile and moment neural networks, mixing quantile and moment features.
result Moment neural network outperforms others for bivariate distributions.
Dense neural networks can't approximate all functions.
problem Approximation capabilities of dense neural networks.
method Model compression approach combining weak regularity lemma and graph neural networks.
result Existence of Lipschitz continuous functions not approximable by dense neural networks.
Review of neural network expressivity and architectures.
problem Understanding neural network expressivity across different architectures.
method Comprehensive overview of approximation results for various neural network types.
result Deep neural networks offer advantages over shallow ones for specific function classes.
Deep neural networks approximate functions in shift-invariant spaces with controlled error.
problem Approximating functions in shift-invariant spaces with neural networks.
method Using deep ReLU neural networks, estimating approximation error bounds based on network width and depth.
result Deep neural networks achieve optimal approximation rates for Sobolev spaces up to a logarithmic factor.
Paper explores neural network approximations on sphere domains.
problem Approximating functionals on sphere domains using neural networks.
method Encoder-decoder framework with spherical harmonics for infinite-dimensional domain.
result Approximation rates of neural networks with different encoder structures.
We extend neural networks with fractional and mixed activation functions for better function approximation.
problem Limitations in approximating higher-order smooth functions in complex spaces.
method Incorporating fractional exponents in activation functions and defining new density functions.
result Improved accuracy and broader applicability of neural network approximation theory.
Deep networks with path norm regularization can approximate analytic functions.
problem Approximating analytic functions with neural networks.
method Path norm regularized deep networks with activation function.
result Deep networks can approximate analytic functions with logarithmic dependence on approximation error.
Neural networks cannot approximate certain functions in Sobolev spaces, leading to unbounded parameter growth.
problem Non-closedness of sets of neural networks in Sobolev spaces.
method Construction of sequences of neural networks whose realizations converge to functions not realizable by neural networks.
result Sets of realized neural networks are not closed in order- ( m − 1 ) (m-1) ( m − 1 ) Sobolev spaces W m − 1 , p W^{m-1,p} W m − 1 , p for p ∈ [ 1 , ∞ ] p \in [1,\infty] p ∈ [ 1 , ∞ ] . Sharp bounds on neural network approximation rates and widths.
problem Estimating approximation rates, metric entropy, and n-widths of shallow neural networks.
method Introducing smoothly parameterized dictionaries and providing upper and lower bounds.
result Sharp bounds on approximation rates, metric entropy, and n-widths for neural networks with various activation functions.
Gradient descent trains shallow neural networks to approximate functions in 1D.
problem Approximating functions in 1D with shallow neural networks trained by gradient descent.
method Gradient descent optimization of non-convex weight space for finite width networks in 1D.
result Gradient descent can approximate functions in 1D with a minimal number of weights, balancing practical performance and theoretical capabilities.
Theory for deep neural network approximation of score function and its derivatives.
problem Handling data distributions with low-dimensional structure and unbounded support.
method Simultaneous approximation of the score function and its derivatives using deep neural networks.
result Approximation error bounds match literature but relax bounded support requirement.
New Zap Q-learning accelerates reinforcement learning with neural networks.
problem Accelerate convergence of reinforcement learning algorithms.
method Introduces a new framework for analysis of stochastic approximation algorithms, proving consistency under non-degeneracy assumption.
result Zap Q-learning with neural network function approximation converges quickly and is robust to function approximation architecture choice.
Complex-valued neural networks can approximate any continuous function.
problem Generalizing the universal approximation theorem to complex-valued networks.
method Characterizing activation functions for complex networks to approximate any continuous function.
result Different activation functions are required for deep vs shallow complex networks to achieve universal approximation.
Paper analyzes Q-learning with neural networks, proving a fast convergence rate.
problem Analyzing the convergence rate of neural Q-learning.
method Finite-time analysis of neural Q-learning with a deep ReLU network.
result Neural Q-learning converges to optimal policy with O ( 1 / T ) O(1/\sqrt{T}) O ( 1/ T ) rate. Optimal rates for shallow ReLU networks in nonparametric regression.
problem Approximating smooth and non-smooth functions with shallow ReLU networks.
method Analysis of shallow ReLU k ^k k neural networks, using variation norms and deep learning theory. result Optimal approximation rates for shallow ReLU networks in nonparametric regression.
New CFNN architecture approximates functions with machine accuracy.
problem Function approximation with high precision.
method Chebyshev Feature Neural Network (CFNN) with learnable frequencies.
result Achieves machine accuracy in function approximation.
We prove that for analytic functions in low dimension, the convergence rate of the deep neural network approximation is exponential.
New bounds on ReLU networks for low-regular functions.
problem Bounding approximation error for ReLU networks on low-regular functions.
method Complexity analysis of Fourier features residual networks to ReLU networks.
result Approximation error bound proportional to target function norm and inversely proportional to network width and depth.
New RBF networks can approximate any continuous function.
problem Approximating any continuous function on a compact subset.
method Replacing smoothing factors with shifts in RBF networks and proving approximation under certain conditions.
result RBF networks can approximate any continuous function on any compact subset.
Deep networks adapt to function regularity and data distribution.
problem Understanding deep learning's adaptability to function regularity and data distribution.
method Developed nonparametric approximation and estimation theories for a broad class of functions using deep ReLU networks.
result Deep neural networks are adaptive to different regularity of functions and nonuniform data distributions.
We study the approximation of measurable functions on the hypercube by functions arising from affine neural networks. Our main achievement is an approximation of any measurable function f : W n → [ − 1 , 1 ] f \colon W_n \to [-1,1] f : W n → [ − 1 , 1 ] up to a prescribed precision ε > 0 \varepsilon>0 ε > 0 by a bounded number of neurons, depending only on ε \varepsilon ε …
Deep, wide ConvResNets can approximate functions and their smoothness.
problem Function approximation and smoothness in deep networks.
method Analyzing ConvResNets, proving their ability to approximate functions and their smoothness.
result Large ConvResNets can approximate functions and exhibit sufficient first-order smoothness.