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Pseudo-orthogonality for graph 1-Laplacian eigenvectors and applications to higher Cheeger constants and data clustering

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Abstract

The data clustering problem consists in dividing a data set into prescribed groups of homogeneous data. This is an NP-hard problem that can be relaxed in the spectral graph theory, where the optimal cuts of a graph are related to the eigenvalues of graph 1-Laplacian. In this paper, we first give new notations to describe the paths, among critical eigenvectors of the graph 1-Laplacian, realizing sets with prescribed genus. We introduce the pseudo-orthogonality to characterize m3(G), a special eigenvalue for the graph 1-Laplacian. Furthermore, we use it to give an upper bound for the third graph Cheeger constant h3(G), that is, h3(G) ⩽ m3(G). This is a first step for proving that the k-th Cheeger constant is the minimum of the 1-Laplacian Raylegh quotient among vectors that are pseudo-orthogonal to the vectors realizing the previous k − 1 Cheeger constants. Eventually, we apply these results to give a method and a numerical algorithm to compute m3 (G), based on a generalized inverse power method.

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Acknowledgements

This work has been partially supported by the MiUR-Dipartimenti di Eccellenza 2018–2022 grant “Sistemi distribuiti intelligenti” of Dipartimento di Ingegneria Elettrica e dell’Informazione “M. Scarano”, by the MiSE-FSC 2014–2020 grant “SUMMa: Smart Urban Mobility Management”, and by GNAMPA of INdAM. The authors would also like to thank D. A. La Manna and V. Mottola for the helpful conversations during the starting stage of this work.

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  1. Dipartimento di Ingegneria Elettrica e dell’Informazione “M. Scarano”, Università degli Studi di Cassino e del Lazio Meridionale, Via G. Di Biasio n. 43, 03043, Cassino (FR), Italy

    Antonio Corbo Esposito & Gianpaolo Piscitelli

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  1. Antonio Corbo Esposito
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  2. Gianpaolo Piscitelli
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Correspondence to Gianpaolo Piscitelli.

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Esposito, A.C., Piscitelli, G. Pseudo-orthogonality for graph 1-Laplacian eigenvectors and applications to higher Cheeger constants and data clustering. Front. Math. China 17, 591–623 (2022). https://doi.org/10.1007/s11464-021-0961-2

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