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On best approximation by multivariate ridge functions with applications to generalized translation networks

arXiv 2412.08453 ↗First posted 11 Dec 2024math.FA cs.LG stat.MLcited by 0

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Abstract (arXiv)

In this paper, we prove sharp upper and lower bounds for the approximation of Sobolev functions by sums of multivariate ridge functions, i.e., for approximation by functions of the form ℝd ni x mapsto sumk=1n varrhok(Ak x) ∈ ℝ with varrhok : ℝℓ → ℝ and Ak ∈ ℝℓ × d. We show that the order of approximation asymptotically behaves as n-r/(d-ℓ), where r is the regularity (order of differentiability) of the Sobolev functions to be approximated. Our lower bound even holds when approximating L^∞-Sobolev functions of regularity r with error measured in L1, while our upper bound applies to the approximation of Lp-Sobolev functions in Lp for any 1 ≤ p ≤ ∞. These bounds generalize well-known results regarding the approximation properties of univariate ridge functions to the multivariate case. We use our results to obtain sharp asymptotic bounds for the approximation of Sobolev functions using generalized translation networks and complex-valued neural networks.

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References · 12

  1. 2017Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networkscited by 311
  2. 2016Benefits of depth in neural networkscited by 250
  3. 2017Optimal approximation of piecewise smooth functions using deep ReLU neural networkscited by 179
  4. 2019Deep Network Approximation Characterized by Number of Neuronscited by 122
  5. 2021Optimal Approximation Rate of ReLU Networks in terms of Width and Depthcited by 96
  6. 2019The phase diagram of approximation rates for deep neural networkscited by 88
  7. 2021A Survey of Complex-Valued Neural Networkscited by 61
  8. 2016Depth-Width Tradeoffs in Approximating Natural Functions with Neural Networkscited by 59
  9. 2021Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neuronscited by 41
  10. 2020The universal approximation theorem for complex-valued neural networkscited by 28
  11. 2023Optimal approximation using complex-valued neural networkscited by 6
  12. 2023Universal approximation with complex-valued deep narrow neural networkscited by 3

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