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A unified framework for harmonic analysis of functions on directed graphs and changing data

Mhaskar, H. N. (2018) A unified framework for harmonic analysis of functions on directed graphs and changing data. Applied and Computational Harmonic Analysis, 44 . pp. 611-644. ISSN 1063-5203.

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We present a general framework for studying harmonic analysis of functions in the settings of various emerging problems in the theory of diffusion geometry. The starting point of the now classical diffusion geometry approach is the construction of a kernel whose discretization leads to an undirected graph structure on an unstructured data set. We study the question of constructing such kernels for directed graph structures, and argue that our construction is essentially the only way to do so using discretizations of kernels. We then use our previous theory to develop harmonic analysis based on the singular value decomposition of the resulting non-self-adjoint operators associated with the directed graph. Next, we consider the question of how functions defined on one space evolve to another space in the paradigm of changing data sets recently introduced by Coifman and Hirn. While the approach of Coifman and Hirn requires that the points on one space should be in a known one-to-one correspondence with the points on the other, our approach allows the identification of only a subset of landmark points. We introduce a new definition of distance between points on two spaces, construct localized kernels based on the two spaces and certain interaction parameters, and study the evolution of smoothness of a function on one space to its lifting to the other space via the landmarks. We develop novel mathematical tools that enable us to study these seemingly different problems in a unified manner.

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Additional Information:© 2016 Elsevier Inc. Received 1 July 2015, Accepted 19 June 2016, Available online 28 June 2016. The research of this author is supported in part by ARO Grant W911NF-15-1-0385.
Funding AgencyGrant Number
Army Research Office (ARO)W911NF-15-1-0385
Subject Keywords:Kernel construction; Directed graphs; Changing data problems; Extension problems; Wavelet-like representation; Tauberian theorem
Record Number:CaltechAUTHORS:20170711-072145868
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Official Citation:H.N. Mhaskar, A unified framework for harmonic analysis of functions on directed graphs and changing data, Applied and Computational Harmonic Analysis, Volume 44, Issue 3, 2018, Pages 611-644, ISSN 1063-5203, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78922
Deposited By: Ruth Sustaita
Deposited On:11 Jul 2017 22:12
Last Modified:07 Mar 2018 17:42

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