Graph Signal Processing for Directed Graphs Based on the Hermitian Laplacian

Furutani, Satoshi; Shibahara, Toshiki; Akiyama, Mitoshi; Hato, Kunio; Aida, Masaki

doi:10.1007/978-3-030-46150-8_27

Cited by 33 publications

(33 citation statements)

References 29 publications

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“…Stated differently, any flow that is not divergence free, i.e., not cyclic, will be penalized by the quadratic form. Since by the fundamental theorem of linear algebra ker(B 1 ) ⊥ im(B 1 ), any such non-cyclic flow can be written as a gradient flow f grad = B 1 v for some vector v of scalar node potentials -in line with the Hodge decomposition discussed in (20).…”

Section: Flow Smoothing and Denoisingmentioning

confidence: 92%

“…For simplicity, we will concentrate on undirected graphs in the ensuing discussion, as this setup can be directly generalized to simplicial complexes. However, graph signal processing may also be considered in the context of directed graphs [20,32].…”

Section: Graph Signals Fourier Transforms and Filtersmentioning

confidence: 99%

“…Clearly, any such flow cannot have a positive net sum along any closed path in the graph and the space orthogonal to im(B 1 ) is thus called the cycle space. As indicated by (20), the cycle space is spanned by two types of cyclic flows. The space im(B 2 ) consists of curl flows that can be composed of linear combinations of local circulations along any 2-simplex (triangular face).…”

Section: The Hodge Laplacian As a Shift Operator For Simplicial Compl...mentioning

confidence: 99%

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Signal processing on simplicial complexes

Seby¹,

Frantzen²,

Roddenberry³

et al. 2021

Preprint

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Higher-order networks have so far been considered primarily in the context of studying the structure of complex systems, i.e., the higher-order or multi-way relations connecting the constituent entities. More recently, a number of studies have considered dynamical processes that explicitly account for such higher-order dependencies, e.g., in the context of epidemic spreading processes or opinion formation. In this chapter, we focus on a closely related, but distinct third perspective: how can we use higher-order relationships to process signals and data supported on higher-order network structures. In particular, we survey how ideas from signal processing of data supported on regular domains, such as time series or images, can be extended to graphs and simplicial complexes. We discuss Fourier analysis, signal denoising, signal interpolation, and nonlinear processing through neural networks based on simplicial complexes. Key to our developments is the Hodge Laplacian matrix, a multi-relational operator that leverages the special structure of simplicial complexes and generalizes desirable properties of the Laplacian matrix in graph signal processing.This chapter is built on the exposition of [46], in particular with respect to the example applications considered. Our discussion here is however more geared towards a reader with a network science background who is less familiar with signal processing. In particular, we provide additional discussion on the relations between filtering and linear dynamical processes on networks.

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Section: Flow Smoothing and Denoisingmentioning

confidence: 92%

Section: Graph Signals Fourier Transforms and Filtersmentioning

confidence: 99%

Section: The Hodge Laplacian As a Shift Operator For Simplicial Compl...mentioning

confidence: 99%

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Signal processing on simplicial complexes

Seby¹,

Frantzen²,

Roddenberry³

et al. 2021

Preprint

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“…First, we consider a small digraph consisting of the 48 contiguous US states with edges going from lower to higher latitude (see Fig. 10) that has been a popular use case in several publications (e.g., [21], [26]). The graph consists 11.…”

Section: A Computing Generalized Boundary Conditionsmentioning

confidence: 99%

“…Another approach is to change the graph shift operator and thus change the underlying definitions of spectrum and Fourier transform. In [26] the Hermitian Laplacian matrix is proposed, which is always diagonalizable. The known directed Laplacian was used in [27] and a scaled version of it, with a detailed study, in [28] but both shifts are not diagonalizable, and our proposed method is applicable in both cases.…”

Section: Introductionmentioning

confidence: 99%

Digraph Signal Processing with Generalized Boundary Conditions

Seifert,

Püschel

2020

Preprint

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Signal processing on directed graphs (digraphs) is problematic, since the graph shift, and thus associated filters, are in general not diagonalizable. Furthermore, the Fourier transform in this case is now obtained from the Jordan decomposition, which may not be computable at all for large graphs. We propose a novel and general solution for this problem based on matrix perturbation theory: We design an algorithm that adds a small number of edges to a given digraph to destroy nontrivial Jordan blocks. The obtained digraph is then diagonalizable and yields, as we show, an approximate eigenbasis and Fourier transform for the original digraph. We explain why and how this construction can be viewed as generalized form of boundary conditions, a common practice in signal processing. Our experiments with random and real world graphs show that we can scale to graphs with a few thousands nodes, and obtain Fourier transforms that are close to orthogonal while still diagonalizing an intuitive notion of convolution. Our method works with adjacency and Laplacian shift and can be used as preprocessing step to enable further processing as we show with a prototypical Wiener filter application.

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