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Partition of Unity Methods for Signal Processing on Graphs

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Abstract

Partition of unity methods (PUMs) on graphs are simple and highly adaptive auxiliary tools for graph signal processing. Based on a greedy-type metric clustering and augmentation scheme, we show how a partition of unity can be generated in an efficient way on graphs. We investigate how PUMs can be combined with a local graph basis function (GBF) approximation method in order to obtain low-cost global interpolation or classification schemes. From a theoretical point of view, we study necessary prerequisites for the partition of unity such that global error estimates of the PUM follow from corresponding local ones. Finally, properties of the PUM as cost-efficiency and approximation accuracy are investigated numerically.

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Acknowledgements

The authors sincerely thank the editors and the anonymous referees for insightful comments and suggestions, which gave the chance to significantly improve the quality of the paper. The first two authors acknowledge support from the Department of Mathematics “Giuseppe Peano” of the University of Torino via projects 2020 “Models and numerical methods in approximation, in applied sciences and in life sciences” and “Mathematical methods in computational sciences”. This research has been accomplished within the RITA “ Research ITalian network on Approximation” and the UMI Group TAA “ Approximation Theory and Applications”. The work was partially supported by INdAM-GNCS, and all the authors are members of the INdAM Research group GNCS.

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Authors and Affiliations

  1. University of Torino, Turin, Italy

    Roberto Cavoretto & Alessandra De Rossi

  2. University of Padova, Padua, Italy

    Wolfgang Erb

  3. Catholic University of Eichstätt-Ingolstadt, Eichstätt, Germany

    Wolfgang Erb

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  1. Roberto Cavoretto
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  2. Alessandra De Rossi
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  3. Wolfgang Erb
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Correspondence to Wolfgang Erb.

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Cavoretto, R., De Rossi, A. & Erb, W. Partition of Unity Methods for Signal Processing on Graphs. J Fourier Anal Appl 27, 66 (2021). https://doi.org/10.1007/s00041-021-09871-w

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