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Representation learning considering high-order relationships in data has recently shown to be advantageous in many applications. The construction of a meaningful hypergraph plays a crucial role in the success of hypergraph-based representation learning methods, which is particularly useful in hypergraph neural networks and hypergraph signal processing. However, a meaningful hypergraph may only be available in specific cases. This paper addresses the challenge of learning the underlying hypergraph topology from the data itself. As in graph signal processing applications, we consider the case in which the data possesses certain regularity or smoothness on the hypergraph. To this end, our method builds on the novel tensor-based hypergraph signal processing framework (t-HGSP) that has recently emerged as a powerful tool for preserving the intrinsic high-order structure of data on hypergraphs. Given the hypergraph spectrum and frequency coefficient definitions within the t-HGSP framework, we propose a method to learn the hypergraph Laplacian from data by minimizing the total variation on the hypergraph (TVL-HGSP). Additionally, we introduce an alternative approach (PDL-HGSP) that improves the connectivity of the learned hypergraph without compromising sparsity and use primal-dual-based algorithms to reduce the computational complexity. Finally, we combine the proposed learning algorithms with novel tensor-based hypergraph convolutional neural networks to propose hypergraph learning-convolutional neural networks (t-HyperGLNN). 

Graph signal processing (GSP) techniques are powerful tools that model complex relationships within large datasets, being now used in a myriad of applications in different areas including data science, communication networks, epidemiology, and sociology. Simple graphs can only model pairwise relationships among data which prevents their application in modeling networks with higher-order relationships. For this reason, some efforts have been made to generalize well-known graph signal processing techniques to more complex graphs such as hypergraphs, which allow capturing higher-order relationships among data. In this article, we provide a new hypergraph signal processing framework (t-HGSP) based on a novel tensor-tensor product algebra that has emerged as a powerful tool for preserving the intrinsic structures of tensors. The proposed framework allows the generalization of traditional GSP techniques while keeping the dimensionality characteristic of the complex systems represented by hypergraphs. To this end, the core elements of the t-HGSP framework are introduced, including the shifting operators and the hypergraph signal. The hypergraph Fourier space is also defined, followed by the concept of bandlimited signals and sampling. In our experiments, we demonstrate the benefits of our approach in applications such as clustering and denoising. 

Compressive spectral imaging reconstruction is performed using smoothness on graphs. In doing so, a highly effective and paralilizable graph-smoothness prior reconstruction algorithm is developed based on simple direct matrix inversion. 

In this work, we introduce the concept of blue noise sampling, traditionally used in imaging applications, for band limited signals on graphs. We show how the spectral and vertex domain characterization of these patterns is connected with results about the quality of the sampling sets already existing in the literature. We provide numerical evidence that shows that these patterns are also competitive with respect to the state of the art sampling techniques in terms of the reconstruction error. 

This paper discusses the generalization of the concept of blue noise sampling from traditional halftoning to signal processing on graphs. Making use of the spatial properties of blue noise, we generate sampling patterns that provide reconstruction errors that are similar to the ones obtained with state of the art approaches. This sampling scheme presents an alternative to those techniques that require spectral decompositions.