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This is a theoretical paper on how well shallow ReLU networks can approximate functions on convex domains, and it shows that the shape of the domain can change the best achievable approximation rate even when dimension, error norm, and weighting are fixed. Its results are mathematical bounds rather than empirical benchmarks, so they clarify limits and trade-offs in neural network approximation rather than offering practical performance gains. For ML readers, the useful takeaway is that domain geometry and coefficient weighting matter for approximation guarantees.
Boundary Geometry Determines Optimal Shallow ReLU Approximation under Weighted Coefficient Constraints
arXiv:2610.36854v1 Announce Type: cross Abstract: We study how boundary geometry affects $L^q$ approximation by shallow ReLU neural networks on convex bodies $K\subset\mathbb R^d$. The target classes are unit balls of weighted variation spaces that can enlarge classical spaces by incorporating domain geometry into constraints on output coefficients. We establish upper and lower bounds for arbitrary convex bodies and determine the optimal polynomial approximation exponents for all planar convex bodies and their products with polytopes, including cubes and circular cylinders. For $C^2$ convex bodies, we determine the optimal exponents in a dimension-dependent range of $q$ extending beyond $2$. When all principal curvatures are positive, the result extends to every finite $q\ge2$. For these classes, the dependence of the optimal exponent on the domain is captured by a single geometric growth exponent associated with small boundary caps. We also identify sharp weight thresholds for retaining the classical approximation exponent and improve previous weights on Euclidean balls while preserving the classical $L^2$ approximation rate. When the target is given as a finite ReLU network, the upper bounds can be realized by a compressed network using only the original neurons, without increasing the weighted coefficient sum. These results show that boundary geometry can change the optimal exponent even when dimension, error norm, and weighting rule are fixed.
Abstract:We study how boundary geometry affects $L^q$ approximation by shallow ReLU neural networks on convex bodies $K\subset\mathbb R^d$. The target classes are unit balls of weighted variation spaces that can enlarge classical spaces by incorporating domain geometry into constraints on output coefficients. We establish upper and lower bounds for arbitrary convex bodies and determine the optimal polynomial approximation exponents for all planar convex bodies and their products with polytopes, including cubes and circular cylinders. For $C^2$ convex bodies, we determine the optimal exponents in a dimension-dependent range of $q$ extending beyond $2$. When all principal curvatures are positive, the result extends to every finite $q\ge2$. For these classes, the dependence of the optimal exponent on the domain is captured by a single geometric growth exponent associated with small boundary caps. We also identify sharp weight thresholds for retaining the classical approximation exponent and improve previous weights on Euclidean balls while preserving the classical $L^2$ approximation rate. When the target is given as a finite ReLU network, the upper bounds can be realized by a compressed network using only the original neurons, without increasing the weighted coefficient sum. These results show that boundary geometry can change the optimal exponent even when dimension, error norm, and weighting rule are fixed.
Machine Learning (stat.ML); Machine Learning (cs.LG); Functional Analysis (math.FA); Metric Geometry (math.MG)
arXiv:2610.36854 [stat.ML]
(or arXiv:2610.36854v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2610.36854 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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