On Kolmogorov's Superposition Theorem and its Applications

We present a Regularization Network approach based on Kolmogorov's superposition theorem (KST) to reconstruct higher dimensional continuous functions from their function values on discrete data points. The ansatz is based on a new constructive proof of a version of the theorem. Additionally, the thesis gives a comprehensive overview on the various versions of KST that exist and its relation to well known approximation schemes and Neural Networks. The efficient representation of higher dimensional continuous functions as superposition of univariate continuous functions suggests the conjecture that in a reconstruction, the exponential dependency of the involved numerical costs on the dimensionality, the so-called curse of dimensionality, can be circumvented. However, this is not the case, since the involved univariate functions are either unknown or not smooth. Therefore, we develop a Regularization Network approach in a reproducing kernel Hilbert space setting such that the restriction of the underlying approximation spaces defines a nonlinear model for function reconstruction. Finally, a verification and analysis of the model is given by various numerical examples.

Autorentext

Jürgen Braun, Dr. rer. nat., studied mathematics with emphasis on scientific computing at the University of Bonn. There, he received his diploma degree in mathematics and doctorate in natural sciences at the Institute for Numerical Simulation. During his postgraduate studies he worked as research assistant.

Klappentext

We present a Regularization Network approach based on Kolmogorov's superposition theorem (KST) to reconstruct higher dimensional continuous functions from their function values on discrete data points. The ansatz is based on a new constructive proof of a version of the theorem. Additionally, the thesis gives a comprehensive overview on the various versions of KST that exist and its relation to well known approximation schemes and Neural Networks. The efficient representation of higher dimensional continuous functions as superposition of univariate continuous functions suggests the conjecture that in a reconstruction, the exponential dependency of the involved numerical costs on the dimensionality, the so-called curse of dimensionality, can be circumvented. However, this is not the case, since the involved univariate functions are either unknown or not smooth. Therefore, we develop a Regularization Network approach in a reproducing kernel Hilbert space setting such that the restriction of the underlying approximation spaces defines a nonlinear model for function reconstruction. Finally, a verification and analysis of the model is given by various numerical examples.