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1. t-HGSP: Hypergraph Signal Processing Using t-Product Tensor Decompositions

   https://doi.org/10.1109/TSIPN.2023.3276687  

   Pena-Pena, Karelia; Lau, Daniel L; Arce, Gonzalo R (January 2023, IEEE Transactions on Signal and Information Processing over Networks)

   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. 

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2. The Dual Graph Shift Operator: Identifying the Support of the Frequency Domain

   https://doi.org/10.1007/s00041-021-09850-1  

   Leus, Geert; Segarra, Santiago; Ribeiro, Alejandro; Marques, Antonio G. (June 2021, Journal of Fourier Analysis and Applications)

   null (Ed.)

   Abstract Contemporary data is often supported by an irregular structure, which can be conveniently captured by a graph. Accounting for this graph support is crucial to analyze the data, leading to an area known as graph signal processing (GSP). The two most important tools in GSP are the graph shift operator (GSO), which is a sparse matrix accounting for the topology of the graph, and the graph Fourier transform (GFT), which maps graph signals into a frequency domain spanned by a number of graph-related Fourier-like basis vectors. This alternative representation of a graph signal is denominated the graph frequency signal. Several attempts have been undertaken in order to interpret the support of this graph frequency signal, but they all resulted in a one-dimensional interpretation. However, if the support of the original signal is captured by a graph, why would the graph frequency signal have a simple one-dimensional support? Departing from existing work, we propose an irregular support for the graph frequency signal, which we coin dual graph. A dual GSO leads to a better interpretation of the graph frequency signal and its domain, helps to understand how the different graph frequencies are related and clustered, enables the development of better graph filters and filter banks, and facilitates the generalization of classical SP results to the graph domain. 

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3. EEG-Based Emotion Classification Using Graph Signal Processing

   https://doi.org/10.1109/ICASSP39728.2021.9414342  

   Saboksayr, Seyed Saman; Mateos, Gonzalo; Cetin, Mujdat (June 2021, 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP))

   null (Ed.)

   The key role of emotions in human life is undeniable. The question of whether there exists a brain pattern associated with a specific emotion is the theme of many affective neuroscience studies. In this work, we bring to bear graph signal processing (GSP) techniques to tackle the problem of automatic emotion recognition using brain signals. GSP is an extension of classical signal processing methods to complex networks where there exists an inherent relation graph. With the help of GSP, we propose a new framework for learning class-specific discriminative graphs. To that end, firstly we assume for each class of observations there exists a latent underlying graph representation. Secondly, we consider the observations are smooth on their corresponding class-specific sough graph while they are non-smooth on other classes’ graphs. The learned class-specific graph-based representations can act as sub-dictionaries and be utilized for the task of emotion classification. Applying the proposed method on an electroencephalogram (EEG) emotion recognition dataset indicates the superiority of our framework over other state-of-the-art methods. 

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4. Body Motion Segmentation via Multilayer Graph Processing for Wearable Sensor Signals

   https://doi.org/10.1109/OJSP.2024.3407662  

   Deng, Qinwen; Zhang, Songyang; Ding, Zhi (January 2024, IEEE Open Journal of Signal Processing)

   Human body motion segmentation plays a major role in many applications, ranging from computer vision to robotics. Among a variety of algorithms, graph-based approaches have demonstrated exciting potential in motion analysis owing to their power to capture the underlying correlations among joints. However, most existing works focus on simpler single-layer geometric structures, whereas multi-layer spatial-temporal graph structure can provide more informative results. To provide an interpretable analysis on multilayer spatial-temporal structures, we revisit the emerging field of multilayer graph signal processing (M-GSP), and propose novel approaches based on M-GSP to human motion segmentation. Specifically, we model the spatial-temporal relationships via multilayer graphs (MLG) and introduce M-GSP spectrum analysis for feature extraction.We present two different M-GSP based algorithms for unsupervised segmentation in the MLG spectrum and vertex domains, respectively. Our experimental results demonstrate the robustness and effectiveness of our proposed methods. 

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5. Protection Against Graph-Based False Data Injection Attacks on Power Systems

   https://doi.org/10.1109/TCNS.2024.3371548  

   Morgenstern, Gal; Kim, Jip; Anderson, James; Zussman, Gil; Routtenberg, Tirza (December 2024, IEEE Transactions on Control of Network Systems)

   Graph signal processing (GSP) has emerged as a powerful tool for practical network applications, including power system monitoring. Recent research has focused on developing GSP-based methods for state estimation, attack detection, and topology identification using the representation of the power system voltages as smooth graph signals. Within this framework, efficient methods have been developed for detecting false data injection (FDI) attacks, which until now were perceived as nonsmooth with respect to the graph Laplacian matrix. Consequently, these methods may not be effective against smooth FDI attacks. In this paper, we propose a graph FDI (GFDI) attack that minimizes the Laplacian-based graph total variation (TV) under practical constraints. We present the GFDI attack as the solution for a non-convex constrained optimization problem. The solution to the GFDI attack problem is obtained through approximating it using ℓ1 relaxation. A series of quadratic programming problems that are classified as convex optimization problems are solved to obtain the final solution. We then propose a protection scheme that identifies the minimal set of measurements necessary to constrain the GFDI output to a high graph TV, thereby enabling its detection by existing GSP-based detectors. Our numerical simulations on the IEEE-57 and IEEE-118 bus test cases reveal the potential threat posed by well-designed GSP-based FDI attacks. Moreover, we demonstrate that integrating the proposed protection design with GSP-based detection can lead to significant hardware cost savings compared to previous designs of protection methods against FDI attacks. 

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