Multiscale wavelets on trees, graphs and high dimensional data: Theory and applications to semi supervised learning

Matan Gavish*, Boaz Nadler, Ronald R. Coifman

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

152 Scopus citations

Abstract

Harmonic analysis, and in particular the relation between function smoothness and approximate sparsity of its wavelet coefficients, has played a key role in signal processing and statistical inference for low dimensional data. In contrast, harmonic analysis has thus far had little impact in modern problems involving high dimensional data, or data encoded as graphs or networks. The main contribution of this paper is the development of a harmonic analysis approach, including both learning algorithms and supporting theory, applicable to these more general settings. Given data (be it high dimensional, graph or network) that is represented by one or more hierarchical trees, we first construct multi-scale wavelet-like orthonormal bases on it. Second, we prove that in analogy to the Euclidean case, function smoothness with respect to a specific metric induced by the tree is equivalent to exponential rate of coefficient decay, that is, to approximate sparsity. These results readily translate to simple practical algorithms for various learning tasks. We present an application to transductive semi-supervised learning.

Original languageAmerican English
Title of host publicationICML 2010 - Proceedings, 27th International Conference on Machine Learning
Pages367-374
Number of pages8
StatePublished - 2010
Externally publishedYes
Event27th International Conference on Machine Learning, ICML 2010 - Haifa, Israel
Duration: 21 Jun 201025 Jun 2010

Publication series

NameICML 2010 - Proceedings, 27th International Conference on Machine Learning

Conference

Conference27th International Conference on Machine Learning, ICML 2010
Country/TerritoryIsrael
CityHaifa
Period21/06/1025/06/10

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