$L^p$ 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… 
[PDF] Semantic Reader
1 Citations

One Citation

Sampling numbers of smoothness classes via $\ell^1$-minimization

,

References

SHOWING 1-10 OF 22 REFERENCES

Optimal Learning

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

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

– 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

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

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

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

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

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

  • A. Barron
  • Computer Science
    Machine 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

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.