Explicit Construction of Approximate Kolmogorov-Arnold Superpositions with C2-Smoothness

Abstract: We explicitly construct an approximate version of the Kolmogorov-Arnold superpositions, which is composed of C2 inner and outer functions, and can approximate an arbitrary alpha-Holder continuous function well. The inner functions are generated by applying suitable translations and dilations to a piecewise C2, strictly increasing function, while the outer functions are constructed row-wise through piecewise C2 interpolation using newly designed shape functions. This novel variant of Kolmogorov-Arnold superpositions overcomes the wild and pathological behaviors of the inherent single variable functions, but retains the essence of Kolmogorov strategy of exact representation, an objective that Sprecher, Neural Networks 144, 2021, has actively pursued.