The Optimal Linear B-splines Approximation via Kolmogorov Superposition Theorem and its Application

We propose a new approach for approximating functions in $C([0,1]^d)$ via Kolmogorov superposition theorem (KST) based on the linear spline interpolation of the outer function in the Kolmogorov representation. We improve the results in \cite{LaiShenKST21} by showing that the optimal rate of approximation based on our proposed approach is $O(\frac{1}{n^2})$, where $n$ denotes the number of knots over $[0,1]$. Furthermore, the approximation constant scales linearly with the dimension $d$. We show that there exists a dense subclass in $C([0,1]^d)$ whose approximation can achieve such optimal rate, and the number of parameters needed in such approximation is at most $O(nd)$. Thus, there is no curse of dimensionality when approximating functions in this subclass. Moreover, for $d\geq 4$, we apply tensor product spline denoising technique to denoise KB-splines and get the smooth LKB-splines. We use LKB-splines as basis to approximate functions for the cases when $d=4$ and $d=6$, which extends the results in \cite{LaiShenKST21}. In addition, we validate via numerical experiments that fewer than $O(nd)$ function values are needed to achieve the rate $O(\frac{1}{n^\beta})$ for some $\beta>0$ based on the smoothness of the outer function. Finally, we demonstrate that our approach can be applied to numerically solving partial differential equation such as the Poisson equation with accurate results.