"Lipschitz Inner Functions in Kolmogorov's Superposition Theorem"
The Kolmogorov Superposition Theorem states that any multivariate continuous real-valued function can be represented exactly as the sum of a small number of compositions of univariate continuous functions. The inner functions involved in this composition suffer from a lack of smoothness; it can be shown that, at best, such functions can be constructed to be Lipschitz with constant 1 and not differentiable. Previous algorithms to compute these inner functions have only been Holder continuous; that these functions fail to be Lipschitz is problematic, as they quickly become computationally too complex to be relevant. In this talk, I will briefly outline the criteria for constructing inner functions, beginning with Kolmogorov's original proof. I will discuss the difficulties in constructing a Lipschitz function that satisfies these criteria, and then motivate an algorithm that overcomes these difficulties and succeeds in such a construction.