边界几何决定浅层 ReLU 网络在加权系数约束下的最优逼近指数
Boundary Geometry Determines Optimal Shallow ReLU Approximation under Weighted Coefficient Constraints
数学味很浓的一篇论文，讲清楚了凸域的边界形状会怎么改变浅层 ReLU 网络的理论逼近速度，做逼近论或网络理论的人可以看看。
论文研究凸体 K⊂R^d 上的 L^q 逼近问题，目标类是加权变差空间的单位球。作者给出了任意凸体的上下界，并确定了所有平面凸体及其与多面体乘积（包括立方体和圆柱体）的最优多项式逼近指数。对 C^2 凸体，在 q 超过 2 的维度相关范围内确定最优指数；当所有主曲率为正时结果扩展到所有有限 q≥2。最优指数对区域的依赖由一个与小边界帽相关的几何增长指数刻画，论文还给出了保持经典逼近指数的锐利权重阈值。
Boundary Geometry Determines Optimal Shallow ReLU Approximation under Weighted Coefficient Constraints
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.