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).

Hypergraph neural networks (HyperGNNs) are a family of deep neural networks designed to perform inference on hypergraphs. HyperGNNs follow either a spectral or a spatial approach, in which a convolution or message-passing operation is conducted based on a hypergraph algebraic descriptor. While many HyperGNNs have been proposed and achieved state-of- the-art performance on broad applications, there have been limited attempts at exploring high dimensional hypergraph descriptors (tensors) and joint node interactions carried by hyperedges. In this paper, we depart from hypergraph matrix representations and present a new tensor-HyperGNN framework (T-HyperGNN) with cross-node interactions. The T-HyperGNN framework consists of T-spectral convolution, T-spatial convo- lution, and T-message-passing HyperGNNs (T-MPHN). The T- spectral convolution HyperGNN is defined under the t-product algebra that closely connects to the spectral space. To im- prove computational efficiency for large hypergraphs, we localize the T-spectral convolution approach to formulate the T-spatial convolution and further devise a novel tensor message-passing algorithm for practical implementation by studying a compressed adjacency tensor representation. Compared to the state-of-the- art approaches, our T-HyperGNNs preserve intrinsic high-order network structures without any hypergraph reduction and model the joint effects of nodes through a cross-node interaction layer. These advantages of our T-HyperGNNs are demonstrated in a wide range of real-world hypergraph datasets.