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

arXiv: https://arxiv.org/abs/2412.08453 · HTML: https://mlsanity.com/paper/2412.08453/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 37 references that are in the map.
- First posted on arXiv 11 Dec 2024.

## Builds on (most cited first, 12 of 12)

- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [Optimal Approximation Rate of ReLU Networks in terms of Width and Depth](https://mlsanity.com/paper/2103.00502/) (2021)
- [The phase diagram of approximation rates for deep neural networks](https://mlsanity.com/paper/1906.09477/) (2019)
- [A Survey of Complex-Valued Neural Networks](https://mlsanity.com/paper/2101.12249/) (2021)
- [Depth-Width Tradeoffs in Approximating Natural Functions with Neural Networks](https://mlsanity.com/paper/1610.09887/) (2016)
- [Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neurons](https://mlsanity.com/paper/2107.02397/) (2021)
- [The universal approximation theorem for complex-valued neural networks](https://mlsanity.com/paper/2012.03351/) (2020)
- [Optimal approximation using complex-valued neural networks](https://mlsanity.com/paper/2303.16813/) (2023)
- [Universal approximation with complex-valued deep narrow neural networks](https://mlsanity.com/paper/2305.16910/) (2023)

## 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.