Exploiting Lipschitz Continuity for the Kolmogorov Superposition Theorem

Event Sponsor: 
Mathmatics and Computer Science Division Seminar - LANS
Start Date: 
Sep 19 2018 - 10:30am
Building/Room: 
Building 240/Room 4301
Location: 
Argonne National Laboratory
Speaker(s): 
Matthew Knepley
Speaker(s) Title: 
University at Buffalo

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.

Miscellaneous Information: 

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September 12, 2018, "Inverse problems in new X-ray imaging techniques: Tools, algorithms and future prospects" Siddharth Maddali Vivekanand, Postdoctoral Appointee, MSD/ANL
September 19, 2018, "Exploiting Lipschitz Continuity for the Kolmogorov Superposition Theorem" Matthew Knepley, Associate Professor, Department of Computer Science and Engineering, University at Buffalo
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