$L^p$ sampling numbers for the Fourier-analytic Barron space
@article{Voigtlaender2022LpSN, title={\$L^p\$ sampling numbers for the Fourier-analytic Barron space}, author={Felix Voigtlaender}, journal={ArXiv}, year={2022}, volume={abs/2208.07605} }
In this paper, we consider Barron functions f : [0 , 1] d → R of smoothness σ > 0, which are functions that can be written as f ( x ) = Z R d F ( ξ ) e 2 πi h x,ξ i dξ with Z R d | F ( ) | · σ < ∞ . For σ = 1, these functions play a prominent role in machine learning, since they can be efficiently approximated by (shallow) neural networks without suffering from the curse of dimensionality. For these functions, we study the following question: Given m point samples f ( x 1 ) , . . . , f ( x m ) of…
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References
SHOWING 1-10 OF 22 REFERENCES
Optimal Learning
- Computer ScienceEncyclopedia of Machine Learning and Data Mining
- 2010
The main results of this paper prove that over-parameterized learning with an appropriate loss function gives a near optimal approximation ˆ f of the function f from which the data is collected.
A note on sampling recovery of multivariate functions in the uniform norm
- MathematicsSIAM J. Numer. Anal.
- 2022
The recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm is studied to obtain preasymptotic estimates for the corresponding sampling numbers and a relation to the corresponding Kolmogorov numbers is pointed out.
Real Analysis
- Mathematics
- 2009
– Weierstrass Theorem Theorem If f is a continuous real-valued function on [a, b] and if any is given, then there exists a polynomial p on [a, b] s.t. |f(x)− p(x)| < for all x ∈ [a, b]. In other…
Random points are optimal for the approximation of Sobolev functions
- MathematicsArXiv
- 2020
The quality of arbitrary sampling points are characterized via the L_\gamma(\Omega)-norm of the distance function $\rm{dist}(\cdot,P)$, which improves upon previous characterizations based on the covering radius of $P.
Nonlinear Approximation by Trigonometric Sums
- Mathematics
- 1995
AbstractWe investigate the $L_p$-error of approximation to a function $f\in L_p({\Bbb T}^d)$ by a linear combination $\sum_{k}c_ke_k$ of $n$
exponentials $e_k(x):= e^{i\langle…
Neural network approximation and estimation of classifiers with classification boundary in a Barron class
- Computer Science, Mathematics
- 2020
The obtained approximation and estimation rates are independent of the dimension of the input, showing that the curse of dimension can be overcome in this setting; in fact, the input dimension only enters in the form of a polynomial factor.
Sharp Bounds on the Approximation Rates, Metric Entropy, and $n$-widths of Shallow Neural Networks
- Computer Science, Mathematics
- 2021
The notion of a smoothly parameterized dictionary is introduced and upper bounds on the non-linear approximation rates, metric entropy and n -widths of variation spaces corresponding to shallow neural networks with a variety of activation functions are given.
Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spaces
- Computer ScienceArXiv
- 2021
This work proves hardness results for the problems of approximation and integration on a novel class of neural network approximation spaces by confirming a conjectured and empirically observed theory-to-practice gap in deep learning and shows that approximation rates of a comparable order of convergence are (at least theoretically) achievable.
Approximation and estimation bounds for artificial neural networks
- Computer ScienceMachine Learning
- 2004
The analysis involves Fourier techniques for the approximation error, metric entropy considerations for the estimation error, and a calculation of the index of resolvability of minimum complexity estimation of the family of networks.
Risk Bounds for High-dimensional Ridge Function Combinations Including Neural Networks
- Computer Science
- 2016
This work bridges non-linear and non-parametric function estimation and includes single-hidden layer nets and shows that the risk is small even when the input dimension of an infinite-dimensional parameterized dictionary is much larger than the available sample size.