Abstract
Signal processing over graphs has recently attracted significant attentions for dealing with structured data. Normal graphs, however, only model pairwise relationships between nodes and are not effective in representing and capturing some high-order relationships of data samples, which are common in many applications such as Internet of Things (IoT). In this work, we propose a new framework of hypergraph signal processing (HGSP) based on tensor representation to generalize the traditional graph signal processing (GSP) to tackle high-order interactions. We introduce the core concepts of HGSP and define the hypergraph Fourier space. We then study the spectrum properties of hypergraph Fourier transform and explain its connection to mainstream digital signal processing. We derive the novel hypergraph sampling theory and present the fundamentals of hypergraph filter design based on the tensor framework. We present HGSP-based methods for several signal processing and data analysis applications. Our experimental results demonstrate significant performance improvement using our HGSP framework over some traditional signal processing solutions.
Highlights
G RAPH-theoretic tools have recently found broad applications in data science owing to their power to model complex relationships in the large structured data sets [1]
We provide an alternative definition of hypergraph Fourier space based on the orthogonal CANDECOMP/PARAFAC (CP) tensor decomposition, together with the corresponding hypergraph Fourier transform (HGFT)
3) We introduce an hypergraph signal processing (HGSP) method for binary classification problems to demonstrate the practical application of HGSP in data analysis
Summary
G RAPH-theoretic tools have recently found broad applications in data science owing to their power to model complex relationships in the large structured data sets [1]. In this article [4], hypergraph signals are associated with each hyperedge, but its framework is limited to cell complexes, which cannot suitably model many real-world data sets and applications. Another shortcoming of the framework in [4] is the lack of detailed analysis and application examples to demonstrate its practicability. To address the aforementioned challenges and generalize the traditional GSP into a more general hypergraph tool to capture high-dimensional interactions, we propose a novel tensorbased HGSP framework in this article.
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