{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,3]],"date-time":"2026-05-03T03:02:57Z","timestamp":1777777377643,"version":"3.51.4"},"reference-count":79,"publisher":"Emerald","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,3,6]]},"abstract":"<jats:p>Graph signal processing (GSP) has seen rapid developmentsin recent years. Since its introduction around ten years ago,we have seen numerous new ideas and practical applicationsrelated to the field. In this tutorial, we give an overviewof some recent advances in generalizing GSP, with a focuson the extension to high-dimensional spaces, models, andstructures. Alongside new frameworks proposed to tacklesuch problems, many new mathematical tools are beingintroduced. In the first part of the monograph, we willreview traditional GSP, highlight the challenges it faces, andmotivate efforts in overcoming such challenges, which willbe the theme of the rest of the monograph.<\/jats:p>","DOI":"10.1561\/2000000119","type":"journal-article","created":{"date-parts":[[2023,3,6]],"date-time":"2023-03-06T05:04:39Z","timestamp":1678079079000},"page":"209-290","source":"Crossref","is-referenced-by-count":5,"title":["Generalizing Graph Signal Processing: High Dimensional Spaces, Models and Structures"],"prefix":"10.1108","volume":"17","author":[{"given":"Xingchao","family":"Jian","sequence":"first","affiliation":[{"name":"Nanyang Technological University ,","place":["Singapore"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Feng","family":"Ji","sequence":"additional","affiliation":[{"name":"Nanyang Technological University ,","place":["Singapore"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wee Peng","family":"Tay","sequence":"additional","affiliation":[{"name":"Nanyang Technological University ,","place":["Singapore"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"140","published-online":{"date-parts":[[2023,3,6]]},"reference":[{"issue":"14","key":"2026040313415331800_ref001","doi-asserted-by":"crossref","first-page":"3775","DOI":"10.1109\/TSP.2016.2546233","article-title":"\u201cEfficient sampling set selection for bandlimited graph signals using graph spectral proxies,\u201d","volume":"64","author":"Anis","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"2026040313415331800_ref002","doi-asserted-by":"crossref","first-page":"2992","DOI":"10.1109\/TSP.2020.2981920","article-title":"\u201cTopological signal processing over simplicial complexes,\u201d","volume":"68","author":"Barbarossa","year":"2020","journal-title":"IEEE T ans. Signal Process."},{"key":"2026040313415331800_ref003","volume-title":"Reproducing kernel Hilbert","author":"Berlinet","year":"2011"},{"key":"2026040313415331800_ref004","article-title":"\u201cWhat arehigher-order networks?\u201d","author":"Bick","year":"2021","journal-title":"arXiv preprint arXiv:2104.11329"},{"key":"2026040313415331800_ref005","first-page":"1026","article-title":"\u201cWeisfeiler and Lehman go topological:Message passing simplicial networks,\u201d","author":"Bodnar","year":"2021","journal-title":"ICML"},{"issue":"1","key":"2026040313415331800_ref006","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1109\/TSP.2017.2755586","article-title":"\u201cGreedy sampling of graphsignals,\u201d","volume":"66","author":"Chamon","year":"2018","journal-title":"IEEE Trans. Signal Process."},{"issue":"24","key":"2026040313415331800_ref007","doi-asserted-by":"crossref","first-page":"6510","DOI":"10.1109\/TSP.2015.2469645","article-title":"\u201cDiscretesignal processing on graphs: Sampling theory,\u201d","volume":"63","author":"Chen","year":"2015","journal-title":"IEEE Trans. Signal"},{"issue":"9","key":"2026040313415331800_ref008","doi-asserted-by":"crossref","first-page":"2320","DOI":"10.1109\/TSP.2019.2904925","article-title":"\u201cAdvances in distributed graphfiltering,\u201d","volume":"67","author":"Coutino","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"2026040313415331800_ref009","first-page":"3887","article-title":"\u201cSelf-driven graph volterra models for higher-order link prediction,\u201d","author":"Coutino","year":"2020","journal-title":"Proc. IEEE Int. Conf. Acoustics, Speech, and Signal"},{"key":"2026040313415331800_ref010","volume-title":"Introduction to Hilbert spaces with","author":"Debnath","year":"2005"},{"issue":"23","key":"2026040313415331800_ref011","doi-asserted-by":"crossref","first-page":"6160","DOI":"10.1109\/TSP.2016.2602809","article-title":"\u201cLearningLaplacian matrix in smooth graph signal representations,\u201d","volume":"64","author":"Dong","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"issue":"6","key":"2026040313415331800_ref012","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1109\/MSP.2020.3014591","article-title":"\u201cGraph signal processing for machine learning: A review and newperspectives,\u201d","volume":"37","author":"Dong","year":"2020","journal-title":"IEEE Signal Process. Mag."},{"key":"2026040313415331800_ref013","article-title":"\u201cSimplicial neuralnetworks,\u201d","author":"Ebli","year":"2020","journal-title":"NeurIPS Workshop in Topological Data Analysis and"},{"key":"2026040313415331800_ref014","first-page":"62","article-title":"\u201cAdaptive graph filters in reproducing kernel Hilbert spaces: Designand performance analysis,\u201d","volume":"7","author":"Elias","year":"2021","journal-title":"IEEE Trans. Signal Inf. Process."},{"key":"2026040313415331800_ref015","doi-asserted-by":"crossref","first-page":"936","DOI":"10.1109\/TSP.2022.3149134","article-title":"\u201cKernel regression over graphs using random Fourier features,\u201d","volume":"70","author":"Elias","year":"2022","journal-title":"IEEE Trans. Signal Process."},{"issue":"24","key":"2026040313415331800_ref016","doi-asserted-by":"crossref","first-page":"6440","DOI":"10.1109\/TSP.2019.2952053","article-title":"\u201cControllability of bandlimited graph processes over random time varying graphs,\u201d","volume":"67","author":"Gama","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"issue":"3","key":"2026040313415331800_ref017","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1109\/TSP.2017.2775589","article-title":"\u201cA time-vertexsignal processing framework: Scalable processing and meaningfulrepresentations for time-series on graphs,\u201d","volume":"66","author":"Grassi","year":"2018","journal-title":"IEEE Trans. Signal"},{"key":"2026040313415331800_ref018","article-title":"\u201cA primer on PAC-Bayesian learning,\u201d","author":"Guedj","year":"2019","journal-title":"arXiv preprint arXiv:1901.05353"},{"key":"2026040313415331800_ref019","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/978-3-030-03574-7_3","volume-title":"Vertex-Frequency Analysis of Graph Signals","author":"Hammond","year":"2019"},{"key":"2026040313415331800_ref020","volume-title":"The Elements of","author":"Hastie","year":"2009"},{"key":"2026040313415331800_ref021","volume-title":"Algebraic topology","author":"Hatcher","year":"2000"},{"key":"2026040313415331800_ref022","doi-asserted-by":"crossref","DOI":"10.1002\/9781118762547","volume-title":"Theoretical Foundations of Functional Data Analysis, With an Introduction to Linear Operators","author":"Hsing","year":"2015"},{"issue":"7","key":"2026040313415331800_ref023","doi-asserted-by":"crossref","first-page":"1189","DOI":"10.1109\/JSTSP.2016.2600859","article-title":"\u201cGraph frequency analysis of brain signals,\u201d","volume":"10","author":"Huang","journal-title":"IEEE J. Sel. Topics Signal Process."},{"issue":"19","key":"2026040313415331800_ref024","doi-asserted-by":"crossref","first-page":"5066","DOI":"10.1109\/TSP.2018.2864654","article-title":"\u201cRating predictionvia graph signal processing,\u201d","volume":"66","author":"Huang","year":"2018","journal-title":"IEEE Trans. Signal Process."},{"key":"2026040313415331800_ref025","volume-title":"Algebra (Graduate Texts in Mathematics) (v. 73)","author":"Hungerford","year":"2002"},{"key":"2026040313415331800_ref026","volume-title":"Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing","author":"Isufi","year":"2017"},{"issue":"2","key":"2026040313415331800_ref027","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1109\/TSP.2016.2614793","article-title":"\u201cAutoregressivemoving average graph filtering,\u201d","volume":"65","author":"Isufi","year":"2017","journal-title":"IEEE Trans. Signal Process."},{"key":"2026040313415331800_ref028","volume-title":"arXiv preprint arXiv:2010.09952","author":"Ji","year":"2021"},{"key":"2026040313415331800_ref029","doi-asserted-by":"crossref","first-page":"41889","DOI":"10.1109\/ACCESS.2022.3167055","article-title":"\u201cSignal processing on simplicialcomplexes with vertex signals,\u201d","volume":"10","author":"Ji","year":"2022","journal-title":"IEEE Access"},{"issue":"10","key":"2026040313415331800_ref030","doi-asserted-by":"crossref","first-page":"2624","DOI":"10.1109\/TSP.2019.2908133","article-title":"\u201cOn the properties of Gromovmatrices and their applications in network inference,\u201d","volume":"67","author":"Ji","year":"2019","journal-title":"IEEE Trans."},{"key":"2026040313415331800_ref031","volume-title":"Proc. IEEE Global Conf. on Signal and Information Processing","author":"Ji","year":"2018"},{"issue":"24","key":"2026040313415331800_ref032","doi-asserted-by":"crossref","first-page":"6188","DOI":"10.1109\/TSP.2019.2952055","article-title":"\u201cA Hilbert space theory of generalized graphsignal processing,\u201d","volume":"67","author":"Ji","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"2026040313415331800_ref033","volume-title":"IEEE Stat. Signal Process. Workshop","author":"Ji","year":"2021"},{"key":"2026040313415331800_ref034","volume-title":"arXiv preprint arXiv:2108.","author":"Ji","year":"2021"},{"key":"2026040313415331800_ref035","first-page":"3414","volume-title":"IEEE Trans. Signal Process.","author":"Jian","year":"2022"},{"issue":"5","key":"2026040313415331800_ref036","doi-asserted-by":"crossref","first-page":"2207","DOI":"10.1109\/TSP.2022.3159393","article-title":"\u201cBayesian estimationof graph signals,\u201d","volume":"70","author":"Kroizer","year":"2022","journal-title":"IEEE Trans. Signal Process."},{"key":"2026040313415331800_ref037","volume-title":"IJCAI-ECAI","author":"Lee","year":"2022"},{"key":"2026040313415331800_ref038","first-page":"5385","volume-title":"Proc. IEEE Int. Conf. Acoustics, Speech, and Signal","author":"Leus"},{"key":"2026040313415331800_ref039","volume-title":"Proc. IEEE Global Conf. on Signal and Information","author":"Loukas","year":"2016"},{"issue":"1","key":"2026040313415331800_ref040","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13634-019-0631-7","article-title":"\u201cStationary time-vertex signal processing,\u201d","volume":"2019","author":"Loukas","year":"2019","journal-title":"EURASIP Journal on Advances in Signal Processing"},{"issue":"11","key":"2026040313415331800_ref041","doi-asserted-by":"crossref","first-page":"1931","DOI":"10.1109\/LSP.2015.2448655","article-title":"\u201cDistributed autoregressivemoving average graph filters,\u201d","volume":"22","author":"Loukas","year":"2015","journal-title":"IEEE Signal Process. Lett."},{"key":"2026040313415331800_ref042","doi-asserted-by":"crossref","DOI":"10.1090\/coll\/060","volume-title":"Large Networks and Graph Limits","author":"Lov\u00e1sz","year":"2012"},{"key":"2026040313415331800_ref043","volume-title":"Proc. Asilomar Conf. on Signals, Systems and","author":"Luo","year":"2012"},{"issue":"22","key":"2026040313415331800_ref044","doi-asserted-by":"crossref","first-page":"5911","DOI":"10.1109\/TSP.2017.2739099","article-title":"\u201cStationarygraph processes and spectral estimation,\u201d","volume":"65","author":"Marques","year":"2017","journal-title":"IEEE Trans. Signal"},{"key":"2026040313415331800_ref045","volume-title":"Chebyshev Polynomials","author":"Mason","year":"2003"},{"issue":"2","key":"2026040313415331800_ref046","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1038\/s41562-017-0260-9","article-title":"\u201cFunctionalalignment with anatomical networks is associated with cognitiveflexibility,\u201d","volume":"2","author":"Medaglia","year":"2018","journal-title":"Nature human behaviour"},{"key":"2026040313415331800_ref047","volume-title":"Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing","author":"Narang","year":"2013"},{"key":"2026040313415331800_ref048","article-title":"\u201cBayesian graph convolutionalneural networks using non-parametric graph learning,\u201d","author":"Pal","year":"2019","journal-title":"ICLR"},{"key":"2026040313415331800_ref049","volume-title":"Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing","author":"Perraudin","year":"2017"},{"issue":"13","key":"2026040313415331800_ref050","doi-asserted-by":"crossref","first-page":"3462","DOI":"10.1109\/TSP.2017.2690388","article-title":"\u201cStationary signal processingon graphs,\u201d","volume":"65","author":"Perraudin","year":"2017","journal-title":"IEEE Trans. Signal Process."},{"key":"2026040313415331800_ref051","first-page":"192","article-title":"\u201cKernel-basedgraph learning from smooth signals: A functional viewpoint,\u201d","volume":"7","author":"Pu","year":"2021","journal-title":"IEEE Trans. Signal Inf. Process. Netw."},{"key":"2026040313415331800_ref052","volume-title":"Advances in Neural Information Processing Systems","author":"Rahimi","year":"2007"},{"key":"2026040313415331800_ref053","first-page":"9020","article-title":"\u201cPrincipled simplicialneural networks for trajectory prediction,\u201d","author":"Roddenberry","year":"2021","journal-title":"ICML"},{"issue":"3","key":"2026040313415331800_ref054","doi-asserted-by":"crossref","first-page":"764","DOI":"10.1109\/TSP.2016.2620116","article-title":"\u201cKernel-based reconstructionof graph signals,\u201d","volume":"65","author":"Romero","year":"2017","journal-title":"IEEE Trans. Signal Process."},{"key":"2026040313415331800_ref055","article-title":"\u201cGraphon neural networksand the transferability of graph neural networks,\u201d","author":"Ruiz","year":"2020","journal-title":"NIPS"},{"key":"2026040313415331800_ref056","volume-title":"Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing","author":"Ruiz","year":"2020"},{"key":"2026040313415331800_ref057","doi-asserted-by":"crossref","first-page":"4961","DOI":"10.1109\/TSP.2021.3106857","article-title":"\u201cGraphon signalprocessing,\u201d","volume":"69","author":"Ruiz","year":"20211","journal-title":"IEEE Trans. Signal Process."},{"issue":"12","key":"2026040313415331800_ref058","doi-asserted-by":"crossref","first-page":"3042","DOI":"10.1109\/TSP.2014.2321121","article-title":"\u201cDiscrete signal processing ongraphs: Frequency analysis,\u201d","volume":"62","author":"Sandryhaila","year":"2014","journal-title":"IEEE Trans. Signal Process."},{"key":"2026040313415331800_ref059","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-030-91374-8_12","volume-title":"Signal Processing on Simplicial Complexes. In: Higher-Order Systems. Understanding Complex Systems","author":"Schaub","year":"2022"},{"key":"2026040313415331800_ref060","first-page":"467","article-title":"\u201cNetwork topology inference from spectral templates,\u201d","volume":"3","author":"Segarra","year":"2017","journal-title":"IEEE Trans. Signal"},{"issue":"15","key":"2026040313415331800_ref061","doi-asserted-by":"crossref","first-page":"4117","DOI":"10.1109\/TSP.2017.2703660","article-title":"\u201cOptimal graph-filter design and applications to distributed linear network operators,\u201d","volume":"65","author":"Segarra","year":"2017","journal-title":"IEEE Trans. Signal Process."},{"issue":"3","key":"2026040313415331800_ref062","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1109\/MSP.2012.2235192","article-title":"\u201cThe emerging field of signal processing on graphs:Extending high-dimensional data analysis to networks and otherirregular domains,\u201d","volume":"30","author":"Shuman","year":"2013","journal-title":"IEEE Signal Process. Mag."},{"issue":"2","key":"2026040313415331800_ref063","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1016\/j.acha.2015.02.005","article-title":"\u201cVertex-frequencyanalysis on graphs,\u201d","volume":"40","author":"Shuman","year":"2016","journal-title":"Applied and Computational Harmonic"},{"issue":"4","key":"2026040313415331800_ref064","first-page":"736","article-title":"\u201cDistributed signal processing via Chebyshev polynomial approximation,\u201d","volume":"4","author":"Shuman","year":"2018","journal-title":"IEEE Trans. Signal Inf. Process. Netw."},{"issue":"6","key":"2026040313415331800_ref065","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1109\/MSP.2020.3013555","article-title":"\u201cMultiway graph signal processing on tensors: Integrative analysis of irregular geometries,\u201d","volume":"37","author":"Stanley","year":"2020","journal-title":"IEEE Signal Process. Mag."},{"key":"2026040313415331800_ref066","doi-asserted-by":"crossref","first-page":"852","DOI":"10.1109\/TIFS.2019.2929446","article-title":"\u201cOn the relationship between inferenceand data privacy in decentralized IoT networks,\u201d","volume":"15","author":"Sun","year":"2020","journal-title":"IEEE Trans. Inf. Forensics Security"},{"issue":"7","key":"2026040313415331800_ref067","doi-asserted-by":"crossref","first-page":"1734","DOI":"10.1109\/TSP.2018.2793871","article-title":"\u201cToward information privacy forthe Internet of Things: A non-parametric learning approach,\u201d","volume":"66","author":"Sun","year":"2018","journal-title":"IEEE Trans. Signal Process."},{"issue":"2","key":"2026040313415331800_ref068","doi-asserted-by":"crossref","first-page":"3035","DOI":"10.1109\/TIFS.2018.2837655","article-title":"\u201cEstimating infection sources in networks using partial timestamps,\u201d","volume":"13","author":"Tang","year":"2018","journal-title":"IEEE Trans. Inf. Forensics"},{"issue":"2","key":"2026040313415331800_ref069","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1109\/JSTSP.2014.2365757","article-title":"\u201cWhose opinion to follow in multihypothesis sociallearning? A large deviations perspective,\u201d","volume":"9","author":"Tay","year":"2015","journal-title":"IEEE J. Sel. Topics Signal Process."},{"issue":"18","key":"2026040313415331800_ref070","doi-asserted-by":"crossref","first-page":"4845","DOI":"10.1109\/TSP.2016.2573748","article-title":"\u201cSignals on graphs:Uncertainty principle and sampling,\u201d","volume":"64","author":"Tsitsvero","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"2026040313415331800_ref071","volume-title":"Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing","author":"Venkitaraman","year":"2020"},{"issue":"4","key":"2026040313415331800_ref072","first-page":"698","article-title":"\u201cPredicting graph signals using kernel regression where the input signal is agnostic to a graph,\u201d","volume":"5","author":"Venkitaraman","year":"2019","journal-title":"IEEE Trans. Signal Inf. Process. Netw."},{"key":"2026040313415331800_ref073","volume-title":"Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics","author":"Wang","year":"2015"},{"key":"2026040313415331800_ref074","first-page":"226","article-title":"\u201cSpectral graph based vertexfrequency wiener filtering for image and graph signal denoising,\u201d","volume":"6","author":"Ya\u011fan","year":"2020","journal-title":"IEEE Trans. Signal Inf. Process. Netw."},{"issue":"3","key":"2026040313415331800_ref075","first-page":"525","article-title":"\u201cUsing social network information in community-based Bayesian truth discovery,\u201d","volume":"5","author":"Yang","year":"2019","journal-title":"IEEE Trans."},{"key":"2026040313415331800_ref076","volume-title":"Functional Analysis","author":"Yosida","year":"1980"},{"key":"2026040313415331800_ref077","article-title":"\u201cOn critical sampling of timevertexgraph signals,\u201d","author":"Yu","year":"2019","journal-title":"Proc. IEEE Global Conf. Signal, Inf. Process."},{"issue":"1","key":"2026040313415331800_ref078","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1109\/JIOT.2019.2950213","article-title":"\u201cIntroducing hypergraph signal processing: Theoretical foundation and practical applications,\u201d","volume":"7","author":"Zhang","year":"2020","journal-title":"IEEE Internet of Things Journal"},{"key":"2026040313415331800_ref079","article-title":"\u201cBayesian graphconvolutional neural networks for semi-supervised classification,\u201d","author":"Zhang","year":"2019","journal-title":"AAAI"}],"container-title":["Foundations and Trends\u00ae in Signal Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.emerald.com\/ftsig\/article-pdf\/17\/3\/209\/11135094\/2000000119en.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/www.emerald.com\/ftsig\/article-pdf\/17\/3\/209\/11135094\/2000000119en.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T18:55:24Z","timestamp":1777488924000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.emerald.com\/ftsig\/article\/17\/3\/209\/1331298\/Generalizing-Graph-Signal-Processing-High"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,6]]},"references-count":79,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,3,6]]}},"URL":"https:\/\/doi.org\/10.1561\/2000000119","relation":{},"ISSN":["1932-8346","1932-8354"],"issn-type":[{"value":"1932-8346","type":"print"},{"value":"1932-8354","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,6]]}}}