In this paper, we prove sharp upper and lower bounds for the approximation of Sobolev functions by sums of multivariate ridge functions, i.e., functions of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathbb {R}}^d \ni x \mapsto \sum _{k=1}^n {\varrho }_k(A_k x) \in {\mathbb {R}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>∋</mo> <mi>x</mi> <mo>↦</mo> <msubsup> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msub> <mi>ϱ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi>k</mi> </msub> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\varrho }_k : {\mathbb {R}}^\ell \rightarrow {\mathbb {R}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ϱ</mi> <mi>k</mi> </msub> <mo>:</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>ℓ</mi> </msup> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$A_k \in {\mathbb {R}}^{\ell \times d}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>k</mi> </msub> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>ℓ</mi> <mo>×</mo> <mi>d</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. We show that the order of approximation asymptotically behaves as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n^{-r/(d-\ell )}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mi>r</mi> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>-</mo> <mi>ℓ</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation>, where r is the regularity (order of differentiability) of the Sobolev functions to be approximated. Our lower bound even holds when approximating <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-Sobolev functions of regularity r with error measured in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>, while our upper bound applies to the approximation of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-Sobolev functions in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$1 \le p \le \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. 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.

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

期刊论文
出版模式：

开放获取
发表日期：

2025年7月1日

查看全文

Paul Geuchen,
Palina Salanevich,
Olov Schavemaker,
Felix Voigtlaender

摘要

In this paper, we prove sharp upper and lower bounds for the approximation of Sobolev functions by sums of multivariate ridge functions, i.e., functions of the form ${\mathbb {R}}^d \ni x \mapsto \sum _{k=1}^n {\varrho }_k(A_k x) \in {\mathbb {R}}$ $R^{d} ∋ x \mapsto \sum_{k = 1}^{n} ϱ_{k} (A_{k} x) \in R$ with ${\varrho }_k : {\mathbb {R}}^\ell \rightarrow {\mathbb {R}}$ $ϱ_{k} : R^{ℓ} \to R$ and $A_k \in {\mathbb {R}}^{\ell \times d}$ $A_{k} \in R^{ℓ \times d}$ . We show that the order of approximation asymptotically behaves as $n^{-r/(d-\ell )}$ $n^{- r / (d - ℓ)}$ , where r is the regularity (order of differentiability) of the Sobolev functions to be approximated. Our lower bound even holds when approximating $L^\infty $ $L^{\infty}$ -Sobolev functions of regularity r with error measured in $L^1$ $L^{1}$ , while our upper bound applies to the approximation of $L^p$ $L^{p}$ -Sobolev functions in $L^p$ $L^{p}$ for any $1 \le p \le \infty $ $1 \leq p \leq \infty$ . 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.