Hypergraph Neural networks (HyperGNNs) and hypergraph signal denoising (HyperGSD) are two fundamental topics in higher-order network modeling. Understanding the connection between these two domains is particularly useful for designing novel HyperGNNs from a HyperGSD perspective, and vice versa. In particular, the tensor-hypergraph convolutional network (T-HGCN) has emerged as a powerful architecture for preserving higher-order interactions on hypergraphs, and this work shows an equivalence relation between a HyperGSD problem and the T-HGCN. Inspired by this intriguing result, we further design a tensor-hypergraph iterative network (T-HGIN) based on the HyperGSD problem, which takes advantage of a multi-step updating scheme in every single layer. Numerical experiments are conducted to show the promising applications of the proposed T-HGIN approach.

Structural health monitoring (SHM) has emerged as a vital field of research, geared towards preserving the longevity and safety of civil infrastructure [1]. A critical component of SHM is the analysis of vibration time series data, which offers insights into the behavior, health, and performance of structures [2]. As infrastructure, especially in urban regions, is subject to a myriad of dynamic forces--ranging from wind to traffic loads - it becomes pivotal to extract clear and meaningful data from the complex vibration signatures that these forces induce. However, one of the significant challenges plaguing SHM practitioners is the interference of noise in these vibration signals, which can distort interpretations and lead to unreliable conclusions. The dynamic response of structures is often manifested as multimodal vibrations, meaning multiple modes or patterns of vibration coexist. These modes, each characterized by its frequency and shape, provide a fingerprint of the structure's health and dynamic properties.

A signal may be defined as any observable change in a quantity over space or time, even if it does not carry information. An analog signal is a continuous stream of values. There are multiple possible values. A digital signal is a discrete stream of values. There are only certain possible values.