We explain how to use Kolmogorov Superposition Theorem (KST) to break the curse of dimensionality when approximating a dense class of multivariate continuous functions. We first show that there is a class of functions called $K$-Lipschitz continuous in $C([0,1]^d)$ which can be approximated by a special ReLU neural network of two hidden layers with a dimension independent approximation rate $O(n^{-1})$ with approximation constant increasing quadratically in $d$. The number of parameters used in such neural network approximation equals to $(6d+2)n$. Next we introduce KB-splines by using linear B-splines to replace the K-outer function and smooth the KB-splines to have the so-called LKB-splines as the basis for approximation. Our numerical evidence shows that the curse of dimensionality is broken in the following sense: When using the standard discrete least squares (DLS) method to approximate a continuous function, there exists a pivotal set of points in $[0,1]^d$ with size at most $O(nd)$ such that the rooted mean squares error (RMSE) from the DLS based on the pivotal set is similar to the RMSE of the DLS based on the original set with size $O(n^d)$. In addition, by using matrix cross approximation technique, the number of LKB-splines used for approximation is the same as the size of the pivotal data set. Therefore, we do not need too many basis functions as well as too many function values to approximate a high dimensional continuous function $f$.

翻译：本文阐释如何利用科尔莫戈罗夫叠加定理（KST）突破高维连续函数稠密类逼近中的维数灾难。首先证明，在$C([0,1]^d)$空间中存在一类称为$K$-利普希茨连续的函数，可通过具有两个隐藏层的特殊ReLU神经网络进行逼近，其逼近率$O(n^{-1})$与维度无关，且逼近常数随$d$呈二次增长。此类神经网络逼近使用的参数数量为$(6d+2)n$。接着引入KB样条，通过用线性B样条替换K-外函数，并对其进行光滑化处理，得到所谓LKB样条作为逼近基函数。数值实验表明，维数灾难在以下意义上被突破：当采用标准离散最小二乘法（DLS）逼近连续函数时，存在一个规模不超过$O(nd)$的$[0,1]^d$中关键点集，使得基于该关键点集的DLS均方根误差（RMSE）与基于规模为$O(n^d)$的原始点集的DLS误差相当。此外，利用矩阵交叉逼近技术，用于逼近的LKB样条数量与关键数据集规模相同。因此，逼近高维连续函数$f$时，无需过多基函数及函数值。