Exploiting Lipschitz Continuity for the Kolmogorov Superposition Theorem

Matthew Knepley

The Curse of Dimensionality constrains computational methods for high-dimensional problems. Many methods to overcome these constraints, including neural networks, projection-pursuit, radial basis functions, and ridge functions, can be explained as approximations of the Kolmogorov Superposition Theorem (KST). This theorem proves the existence of a representation of a multivariate continuous function as the superposition of a small number of univariate functions. These univariate functions are defined on a series of grids that are refined during the construction process. Unfortunately, KST is dif ficult to use directly because the resulting representation is highly nonsmooth. At best, the functions involved in the superpositions are Lipschitz continuous and not differentiable, and the best known constructions are merely Holder continuous. We describe the first known algorithm to construct a Lipschitz KST inner function. The resulting inner function induced by the spatial decomposition is independent of the multivariate function being represented, depending only on the spatial dimensions of the domain. This construction could potentially be the basis for understanding compact approximations of high-dimensional functions.