The Kolmogorov Superposition Theorem can Break the Curse of Dimensionality When Approximating High Dimensional Functions

We explain how to use Kolmogorov Superposition Theorem (KST) to break the curse of dimension when approximating continuous multivariate functions. We first show that there is a class of functions called $K$-Lipschitz continuous and can be approximated by a ReLU neural network of two layers to have an approximation order $O(d^2/n)$, and then we introduce the K-modulus of continuity of multivariate functions and derive the approximation rate for any continuous function $f\in C([0,1]^d)$ based on KST. Next we introduce KB-splines by using uniform B-splines to replace the K-outer function and their smooth version called LKB-splines to approximate high dimensional functions. Our numerical evidence shows that the curse of dimension is broken in the following sense. When using the standard discrete least squares method (DLS) to approximate a continuous function $f$ over $[0, 1]^d$, one expects to use a dense data set $P$, a large number of data locations and function values over the locations. Based on the LKB splines, the structure of K-inner functions leads to a sparse solution of the linear system associated with the DLS. Furthermore, there exists a magic set of points from $P$ with much small in size such that the rooted mean squares error (RMSE) from the DLS based on the magic set is similar to the RMSE of the DLS based on the original set $P$. In addition, the number of LKB-splines used for approximation is the same as the size of the magic data set. Hence we not need a lot of basis functions and a lot of data locations and function values to approximate a high dimensional continuous function $f$ when $f$ is not very oscillated.