{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T17:33:38Z","timestamp":1781112818037,"version":"3.54.1"},"reference-count":68,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,4,20]],"date-time":"2021-04-20T00:00:00Z","timestamp":1618876800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["678304"],"award-info":[{"award-number":["678304"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Horizon 2020","award":["666992"],"award-info":[{"award-number":["666992"]}]},{"name":"Horizon 2020","award":["826421"],"award-info":[{"award-number":["826421"]}]},{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-10-IAIHU-06"],"award-info":[{"award-number":["ANR-10-IAIHU-06"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-19-P3IA-0001"],"award-info":[{"award-number":["ANR-19-P3IA-0001"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Network analysis provides a rich framework to model complex phenomena, such as human brain connectivity. It has proven efficient to understand their natural properties and design predictive models. In this paper, we study the variability within groups of networks, i.e., the structure of connection similarities and differences across a set of networks. We propose a statistical framework to model these variations based on manifold-valued latent factors. Each network adjacency matrix is decomposed as a weighted sum of matrix patterns with rank one. Each pattern is described as a random perturbation of a dictionary element. As a hierarchical statistical model, it enables the analysis of heterogeneous populations of adjacency matrices using mixtures. Our framework can also be used to infer the weight of missing edges. We estimate the parameters of the model using an Expectation-Maximization-based algorithm. Experimenting on synthetic data, we show that the algorithm is able to accurately estimate the latent structure in both low and high dimensions. We apply our model on a large data set of functional brain connectivity matrices from the UK Biobank. Our results suggest that the proposed model accurately describes the complex variability in the data set with a small number of degrees of freedom.<\/jats:p>","DOI":"10.3390\/e23040490","type":"journal-article","created":{"date-parts":[[2021,4,20]],"date-time":"2021-04-20T13:58:04Z","timestamp":1618927084000},"page":"490","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Understanding the Variability in Graph Data Sets through Statistical Modeling on the Stiefel Manifold"],"prefix":"10.3390","volume":"23","author":[{"given":"Cl\u00e9ment","family":"Mantoux","sequence":"first","affiliation":[{"name":"ARAMIS Project Team, Inria, 75013 Paris, France"},{"name":"ARAMIS Lab, Brain and Spine Institute, ICM, INSERM UMR 1127, CNRS UMR 7225, Sorbonne University, H\u00f4pital de la Piti\u00e9-Salp\u00eatri\u00e8re, 75013 Paris, France"},{"name":"CMAP, \u00c9cole Polytechnique, 91120 Palaiseau, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Baptiste","family":"Couvy-Duchesne","sequence":"additional","affiliation":[{"name":"ARAMIS Project Team, Inria, 75013 Paris, France"},{"name":"ARAMIS Lab, Brain and Spine Institute, ICM, INSERM UMR 1127, CNRS UMR 7225, Sorbonne University, H\u00f4pital de la Piti\u00e9-Salp\u00eatri\u00e8re, 75013 Paris, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2220-0567","authenticated-orcid":false,"given":"Federica","family":"Cacciamani","sequence":"additional","affiliation":[{"name":"ARAMIS Project Team, Inria, 75013 Paris, France"},{"name":"ARAMIS Lab, Brain and Spine Institute, ICM, INSERM UMR 1127, CNRS UMR 7225, Sorbonne University, H\u00f4pital de la Piti\u00e9-Salp\u00eatri\u00e8re, 75013 Paris, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"St\u00e9phane","family":"Epelbaum","sequence":"additional","affiliation":[{"name":"ARAMIS Project Team, Inria, 75013 Paris, France"},{"name":"ARAMIS Lab, Brain and Spine Institute, ICM, INSERM UMR 1127, CNRS UMR 7225, Sorbonne University, H\u00f4pital de la Piti\u00e9-Salp\u00eatri\u00e8re, 75013 Paris, France"},{"name":"Institute of Memory and Alzheimer\u2019s Disease (IM2A), Centre of Excellence of Neurodegenerative Disease (CoEN), CIC Neurosciences, AP-HP, Department of Neurology, H\u00f4pital de la Piti\u00e9-Salp\u00eatri\u00e8re, 75013 Paris, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stanley","family":"Durrleman","sequence":"additional","affiliation":[{"name":"ARAMIS Project Team, Inria, 75013 Paris, France"},{"name":"ARAMIS Lab, Brain and Spine Institute, ICM, INSERM UMR 1127, CNRS UMR 7225, Sorbonne University, H\u00f4pital de la Piti\u00e9-Salp\u00eatri\u00e8re, 75013 Paris, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"St\u00e9phanie","family":"Allassonni\u00e8re","sequence":"additional","affiliation":[{"name":"Centre de Recherche des Cordeliers, Universit\u00e9 de Paris, INSERM UMR 1138, Sorbonne Universit\u00e9, 75006 Paris, France"},{"name":"HEKA Project Team, Inria, 75006 Paris, France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,20]]},"reference":[{"key":"ref_1","unstructured":"Newman, M.E.J. (2012). Networks\u2014An Introduction, Oxford University Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"9984","DOI":"10.1038\/s41598-019-46380-9","article-title":"Community Detection on Networks with Ricci Flow","volume":"9","author":"Ni","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_3","first-page":"1","article-title":"A Survey of Link Prediction in Complex Networks","volume":"49","author":"Berzal","year":"2016","journal-title":"ACM Comput. Surv."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1038\/nprot.2016.178","article-title":"Using Connectome-Based Predictive Modeling to Predict Individual Behavior from Brain Connectivity","volume":"12","author":"Shen","year":"2017","journal-title":"Nat. Protoc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/BF01201026","article-title":"Metric Inference for Social Networks","volume":"11","author":"Banks","year":"1994","journal-title":"J. Classif."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1059","DOI":"10.1016\/j.neuroimage.2009.10.003","article-title":"Complex Network Measures of Brain Connectivity: Uses and Interpretations","volume":"52","author":"Rubinov","year":"2010","journal-title":"NeuroImage"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"K\u016frkov\u00e1, V., Manolopoulos, Y., Hammer, B., Iliadis, L., and Maglogiannis, I. (2018). GraphVAE: Towards Generation of Small Graphs Using Variational Autoencoders. Artificial Neural Networks and Machine Learning\u2014ICANN 2018, Springer International Publishing. Lecture Notes in Computer Science.","DOI":"10.1007\/978-3-030-01424-7"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Pozzi, F.A., Fersini, E., Messina, E., and Liu, B. (2016). Sentiment Analysis in Social Networks, Morgan Kaufmann.","DOI":"10.1016\/B978-0-12-804412-4.00001-2"},{"key":"ref_9","first-page":"3697","article-title":"Geometric Matrix Completion with Recurrent Multi-Graph Neural Networks","volume":"30","author":"Monti","year":"2017","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Narayanan, T., and Subramaniam, S. (2013). Community Structure Analysis of Gene Interaction Networks in Duchenne Muscular Dystrophy. PLoS ONE, 8.","DOI":"10.1371\/journal.pone.0067237"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1097\/WCO.0b013e32833aa567","article-title":"Graph Theoretical Modeling of Brain Connectivity","volume":"23","author":"He","year":"2010","journal-title":"Curr. Opin. Neurol."},{"key":"ref_12","first-page":"2224","article-title":"Convolutional Networks on Graphs for Learning Molecular Fingerprints","volume":"28","author":"Duvenaud","year":"2015","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Lov\u00e1sz, L. (2012). Large Networks and Graph Limits, American Mathematical Society. Colloquium Publications.","DOI":"10.1090\/coll\/060"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1214\/09-EJS548","article-title":"Discrete Temporal Models of Social Networks","volume":"4","author":"Hanneke","year":"2010","journal-title":"Electron. J. Stat."},{"key":"ref_15","unstructured":"Fornito, A., Zalesky, A., and Bullmore, E. (2016). Fundamentals of Brain Network Analysis, Academic Press."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Zheng, W., Yao, Z., Li, Y., Zhang, Y., Hu, B., Wu, D., and Alzheimer\u2019s Disease Neuroimaging Initiative (2019). Brain Connectivity Based Prediction of Alzheimer\u2019s Disease in Patients With Mild Cognitive Impairment Based on Multi-Modal Images. Front. Hum. Neurosci., 13.","DOI":"10.3389\/fnhum.2019.00399"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.cosrev.2017.11.002","article-title":"The Journey of Graph Kernels through Two Decades","volume":"27","author":"Ghosh","year":"2018","journal-title":"Comput. Sci. Rev."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.neuroimage.2017.01.077","article-title":"Effects of Aging on Functional and Structural Brain Connectivity","volume":"160","author":"Damoiseaux","year":"2017","journal-title":"NeuroImage"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Chikuse, Y. (2003). Statistics on Special Manifolds, Springer. Lecture Notes in Statistics.","DOI":"10.1007\/978-0-387-21540-2"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Harris, J.K. (2014). An Introduction to Exponential Random Graph Modeling, SAGE. Number 173 in Quantitative Applications in the Social Sciences.","DOI":"10.4135\/9781452270135"},{"key":"ref_21","first-page":"290","article-title":"On Random Graphs","volume":"6","author":"Erdos","year":"1959","journal-title":"Publ. Math."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Doreian, P., Batagelj, V., and Ferligoj, A. (2020). Bayesian Stochastic Blockmodeling. Advances in Network Clustering and Blockmodeling, Wiley. Available online: http:\/\/arxiv.org\/abs\/1705.10225.","DOI":"10.1002\/9781119483298"},{"key":"ref_23","unstructured":"Chandna, S., and Maugis, P.A. (2020). Nonparametric Regression for Multiple Heterogeneous Networks. arXiv."},{"key":"ref_24","unstructured":"Zhang, Z., Cui, P., and Zhu, W. (2020). Deep Learning on Graphs: A Survey. arXiv."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Schirmer, M.D., Venkataraman, A., Rekik, I., Kim, M., and Chung, A.W. (2019). Adversarial Connectome Embedding for Mild Cognitive Impairment Identification Using Cortical Morphological Networks. Connectomics in NeuroImaging, Springer International Publishing. Lecture Notes in Computer Science.","DOI":"10.1007\/978-3-030-32391-2"},{"key":"ref_26","first-page":"387","article-title":"Attention-Guided Deep Graph Neural Network for Longitudinal Alzheimer\u2019s Disease Analysis","volume":"Volume 12267","author":"Martel","year":"2020","journal-title":"Medical Image Computing and Computer Assisted Intervention \u2014MICCAI 2020"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1214\/10-AOAS403","article-title":"A Mixed Effects Model for Longitudinal Relational and Network Data, with Applications to International Trade and Conflict","volume":"5","author":"Westveld","year":"2011","journal-title":"Ann. Appl. Stat."},{"key":"ref_28","first-page":"709","article-title":"Integrating Neural Networks and Dictionary Learning for Multidimensional Clinical Characterizations from Functional Connectomics Data","volume":"Volume 11766","author":"Shen","year":"2019","journal-title":"Medical Image Computing and Computer Assisted Intervention\u2014MICCAI 2019"},{"key":"ref_29","unstructured":"Liu, M., Zhang, Z., and Dunson, D.B. (2019). Auto-Encoding Graph-Valued Data with Applications to Brain Connectomes. arXiv."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1137\/S0895479895290954","article-title":"The Geometry of Algorithms with Orthogonality Constraints","volume":"20","author":"Edelman","year":"1998","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1137\/16M1074485","article-title":"A Matrix-Algebraic Algorithm for the Riemannian Logarithm on the Stiefel Manifold under the Canonical Metric","volume":"38","author":"Zimmermann","year":"2017","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1111\/j.2517-6161.1977.tb01610.x","article-title":"The von Mises\u2013Fisher Matrix Distribution in Orientation Statistics","volume":"39","author":"Khatri","year":"1977","journal-title":"J. R. Stat. Soc. Ser. B (Methodol.)"},{"key":"ref_33","unstructured":"Karush, W. (1939). Minima of Functions of Several Variables with Inequalities as Side Constraints. [Masters\u2019s Thesis, Department of Mathematics, University of Chicago]."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1051\/ps:2004007","article-title":"Coupling a Stochastic Approximation Version of EM with an MCMC Procedure","volume":"8","author":"Kuhn","year":"2004","journal-title":"ESAIM Probab. Stat."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1093\/pan\/mph012","article-title":"Modeling Dependencies in International Relations Networks","volume":"12","author":"Hoff","year":"2004","journal-title":"Political Anal."},{"key":"ref_36","unstructured":"Hoff, P.D. (2007). Modeling Homophily and Stochastic Equivalence in Symmetric Relational Data. Proceedings of the 20th International Conference on Neural Information Processing Systems, Curran Associates Inc.. NIPS\u201907."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"674","DOI":"10.1198\/016214506000001310","article-title":"Model Averaging and Dimension Selection for the Singular Value Decomposition","volume":"102","author":"Hoff","year":"2007","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1198\/jcgs.2009.07177","article-title":"Simulation of the Matrix Bingham\u2014von Mises\u2014Fisher Distribution, With Applications to Multivariate and Relational Data","volume":"18","author":"Hoff","year":"2009","journal-title":"J. Comput. Graph. Stat."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Martel, A.L., Abolmaesumi, P., Stoyanov, D., Mateus, D., Zuluaga, M.A., Zhou, S.K., Racoceanu, D., and Joskowicz, L. (2020). Estimating Common Harmonic Waves of Brain Networks on Stiefel Manifold. Medical Image Computing and Computer Assisted Intervention\u2014MICCAI 2020, Springer International Publishing. Lecture Notes in Computer Science.","DOI":"10.1007\/978-3-030-59716-0"},{"key":"ref_40","first-page":"125","article-title":"A Stochastic Algorithm for Probabilistic Independent Component Analysis","volume":"6","author":"Younes","year":"2012","journal-title":"Ann. Appl. Stat."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.2517-6161.1977.tb01600.x","article-title":"Maximum Likelihood from Incomplete Data Via the EM Algorithm","volume":"39","author":"Dempster","year":"1977","journal-title":"J. R. Stat. Soc. Ser. B (Methodol.)"},{"key":"ref_42","first-page":"641","article-title":"Construction of Bayesian Deformable Models via a Stochastic Approximation Algorithm: A Convergence Study","volume":"16","author":"Kuhn","year":"2010","journal-title":"Bernoulli"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1583","DOI":"10.1214\/21-EJS1827","article-title":"On the Convergence of Stochastic Approximations under a Subgeometric Ergodic Markov Dynamic","volume":"15","author":"Debavelaere","year":"2021","journal-title":"Electron. J. Stat."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1093\/biomet\/57.1.97","article-title":"Monte Carlo Sampling Methods Using Markov Chains and Their Applications","volume":"57","author":"Hastings","year":"1970","journal-title":"Biometrika"},{"key":"ref_45","unstructured":"Robert, C.P., and Casella, G. (2010). Monte Carlo Statistical Methods, Springer."},{"key":"ref_46","unstructured":"Li, J., Fuxin, L., and Todorovic, S. (2020). Efficient Riemannian Optimization On The Stiefel Manifold Via The Cayley Transform. arXiv."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"871","DOI":"10.1214\/19-BA1176","article-title":"Conjugate Priors and Posterior Inference for the Matrix Langevin Distribution on the Stiefel Manifold","volume":"15","author":"Pal","year":"2020","journal-title":"Bayesian Anal."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1214\/aos\/1176344681","article-title":"Maximum Likelihood Estimators for the Matrix Von Mises-Fisher and Bingham Distributions","volume":"7","author":"Jupp","year":"1979","journal-title":"Ann. Stat."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1093\/biomet\/ast021","article-title":"Saddlepoint Approximations for the Normalizing Constant of Fisher\u2013Bingham Distributions on Products of Spheres and Stiefel Manifolds","volume":"100","author":"Kume","year":"2013","journal-title":"Biometrika"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.neucom.2018.01.048","article-title":"Classification of Matrix-Variate Fisher\u2013Bingham Distribution via Maximum Likelihood Estimation Using Manifold Valued Data","volume":"295","author":"Ali","year":"2018","journal-title":"Neurocomputing"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Butler, R.W. (2007). Saddlepoint Approximations with Applications, Cambridge University Press.","DOI":"10.1017\/CBO9780511619083"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"2794","DOI":"10.1007\/s11263-020-01337-8","article-title":"Learning the Clustering of Longitudinal Shape Data Sets into a Mixture of Independent or Branching Trajectories","volume":"128","author":"Debavelaere","year":"2020","journal-title":"Int. J. Comput. Vis."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Lam, S.K., Pitrou, A., and Seibert, S. (2015). Numba: A LLVM-Based Python JIT Compiler. Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC, Association for Computing Machinery. LLVM \u201915.","DOI":"10.1145\/2833157.2833162"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"883","DOI":"10.1109\/TSP.2012.2226167","article-title":"Empirical Arithmetic Averaging Over the Compact Stiefel Manifold","volume":"61","author":"Kaneko","year":"2013","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1150","DOI":"10.1016\/j.physa.2010.11.027","article-title":"Link Prediction in Complex Networks: A Survey","volume":"390","author":"Lu","year":"2011","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_56","first-page":"5171","article-title":"Link Prediction Based on Graph Neural Networks","volume":"31","author":"Zhang","year":"2018","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Fr\u00fchwirth-Schnatter, S., Celeux, G., and Robert, C.P. (2019). Model Selection for Mixture Models\u2014Perspectives and Strategies. Handbook of Mixture Analysis, CRC Press. [1st ed.].","DOI":"10.1201\/9780429055911"},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Sudlow, C., Gallacher, J., Allen, N., Beral, V., Burton, P., Danesh, J., Downey, P., Elliott, P., Green, J., and Landray, M. (2015). UK Biobank: An Open Access Resource for Identifying the Causes of a Wide Range of Complex Diseases of Middle and Old Age. PLoS Med., 12.","DOI":"10.1371\/journal.pmed.1001779"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1016\/S1053-8119(03)00097-1","article-title":"Independent Component Analysis of Nondeterministic fMRI Signal Sources","volume":"19","author":"Kiviniemi","year":"2003","journal-title":"NeuroImage"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1016\/j.neuroimage.2013.09.069","article-title":"The Structural\u2013Functional Connectome and the Default Mode Network of the Human Brain","volume":"102","author":"Horn","year":"2014","journal-title":"NeuroImage"},{"key":"ref_61","unstructured":"Eysenck, M.W. (2010). Cognitive Psychology: A Student\u2019s Handbook, Psychology Press."},{"key":"ref_62","unstructured":"Purves, D., Augustine, G.J., Fitzpatrick, D., Hall, W.C., LaMantia, A.S., Mooney, R.D., Platt, M.L., and White, L.E. (2017). Neuroscience, Sinauer Associates is an Imprint of Oxford University Press. [6th ed.]."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"94215","DOI":"10.1109\/ACCESS.2019.2928130","article-title":"Low-Rank Matrix Completion: A Contemporary Survey","volume":"7","author":"Nguyen","year":"2019","journal-title":"IEEE Access"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"835","DOI":"10.1007\/s11222-017-9765-3","article-title":"On the Exact Maximum Likelihood Inference of Fisher\u2013Bingham Distributions Using an Adjusted Holonomic Gradient Method","volume":"28","author":"Kume","year":"2018","journal-title":"Stat. Comput."},{"key":"ref_65","first-page":"2404","article-title":"Learning Spatiotemporal Trajectories from Manifold-Valued Longitudinal Data","volume":"28","author":"Schiratti","year":"2015","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.acha.2010.04.005","article-title":"Wavelets on Graphs via Spectral Graph Theory","volume":"30","author":"Hammond","year":"2011","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref_67","doi-asserted-by":"crossref","unstructured":"Atasoy, S., Donnelly, I., and Pearson, J. (2016). Human Brain Networks Function in Connectome-Specific Harmonic Waves. Nat. Commun., 7.","DOI":"10.1038\/ncomms10340"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/S0169-7439(01)00155-1","article-title":"PLS-Regression: A Basic Tool of Chemometrics","volume":"58","author":"Wold","year":"2001","journal-title":"Chemom. Intell. Lab. Syst."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/4\/490\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:50:12Z","timestamp":1760161812000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/4\/490"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,20]]},"references-count":68,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,4]]}},"alternative-id":["e23040490"],"URL":"https:\/\/doi.org\/10.3390\/e23040490","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,20]]}}}