{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:11:30Z","timestamp":1760242290681,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2017,3,17]],"date-time":"2017-03-17T00:00:00Z","timestamp":1489708800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the National Natural Science Foundations of China","award":["61179031"],"award-info":[{"award-number":["61179031"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, firstly, manifoldPD(n)consisting of alln\u00d7nsymmetric positive-definite matrices is introduced based on matrix information geometry; Secondly, the geometrical structures of information submanifold ofPD(n)are presented including metric, geodesic and geodesic distance; Thirdly, the information resolution with sensor networks is presented by three classical measurement models based on information submanifold; Finally, the bearing-only tracking by single sensor is introduced by the Fisher information matrix. The preliminary analysis results introduced in this paper indicate that information submanifold is able to offer consistent and more comprehensive means to understand and solve sensor network problems for targets resolution and tracking, which are not easily handled by some conventional analysis methods.<\/jats:p>","DOI":"10.3390\/e19030131","type":"journal-article","created":{"date-parts":[[2017,3,17]],"date-time":"2017-03-17T11:22:56Z","timestamp":1489749776000},"page":"131","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Information Submanifold Based on SPD Matrices and Its Applications to Sensor Networks"],"prefix":"10.3390","volume":"19","author":[{"given":"Hao","family":"Xu","sequence":"first","affiliation":[{"name":"The School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huafei","family":"Sun","sequence":"additional","affiliation":[{"name":"The School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China"},{"name":"Beijing Key Laboratory on MCAACI, Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aung","family":"Win","sequence":"additional","affiliation":[{"name":"The School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,3,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/s12559-009-9026-7","article-title":"Learning the Fre\u2019chet Mean over the Manifold of Symmetric Positive-Definite Matrices","volume":"1","author":"Fiori","year":"2009","journal-title":"Cognit. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Harandi, M.T., Salzmann, M., and Hartley, R. (2014). From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices, Springer.","DOI":"10.1007\/978-3-319-10605-2_2"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1137\/050637996","article-title":"Geometric Means in a Novel Vector Space Structures on Symmetrics Positive-Definite Matrices","volume":"29","author":"Arsigny","year":"2007","journal-title":"Siam J. Matrix Anal. Appl."},{"key":"ref_4","first-page":"1","article-title":"Diffusion MNR Imaging","volume":"7","year":"1991","journal-title":"Magn. Reson. Q."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1016\/j.neuroimage.2006.09.027","article-title":"Measuring Brain Variability by Extrapolating Sparse Tensor Fields Measured on Sulcal Lines","volume":"34","author":"Fillard","year":"2007","journal-title":"Neuroimage"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1007\/978-3-642-33709-3_16","article-title":"Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approach","volume":"7573","author":"Harandi","year":"2012","journal-title":"Lect. Notes Comput. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Jayasumana, S., Hartley, R., Salzmann, M., Li, H., and Harandi, M. (2013, January 23\u201328). Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices. Proceedings of the 2013 IEEE Conference on Computer Vision and Pattern Recognition, Portland, OR, USA.","DOI":"10.1109\/CVPR.2013.17"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1137\/S0895479803436937","article-title":"A Differential Geometric Approach to The Geometric Mean of Symmetric Positive-Definite Matrices","volume":"26","author":"Moakher","year":"2005","journal-title":"Siam J. Matrix Anal. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Hall, B. (2003). Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer.","DOI":"10.1007\/978-0-387-21554-9"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/0893-9659(94)00112-P","article-title":"Statistical Test Based on Geodesic Distances","volume":"8","author":"Menendez","year":"1995","journal-title":"Appl. Math. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1625","DOI":"10.1162\/089976601750265036","article-title":"A Theory for Learning by Weight Flow on Stiefel-Grassman Manifold","volume":"13","author":"Fiori","year":"2001","journal-title":"Neural Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"938","DOI":"10.1137\/100817115","article-title":"Solving Minimal-Distance Problems over the Manifold of Real-Symplectic Matrices","volume":"32","author":"Fiori","year":"2011","journal-title":"Siam J. Matrix Anal. Appl."},{"key":"ref_13","first-page":"341","article-title":"Computing the Karcher Mean of Symmetric Positive Definite Matrices","volume":"4","author":"Bini","year":"2007","journal-title":"Stat. Methodol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1007\/978-3-642-00826-9_6","article-title":"Interactions Between Symmetric Cones and Information Geometrics: Bruhat-Tits and Siegel Spaces Models for High Resolution Autregressive Doppler Imagery","volume":"5416","author":"Barbaresco","year":"2009","journal-title":"Springer Lect. Notes Comput. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Barbaresco, F. (2008, January 26\u201330). Innovative Tools for Radar Signal Processing based on Cartan\u2019s Geometry of SPD Matrices and Information Geometry. Proceedings of the IEEE Radar Conference, Rome, Italy.","DOI":"10.1109\/RADAR.2008.4720937"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Nielsen, F., and Bhatia, R. (2013). Matrix Information Geometry, Springer.","DOI":"10.1007\/978-3-642-30232-9"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.automatica.2016.02.026","article-title":"Information Matrix and D-optimal Design with Gaussian Inputs for Wiener Model Identification","volume":"69","author":"Mahata","year":"2016","journal-title":"Automatica"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1109\/78.890346","article-title":"Parameter Estimation Problems with Singular Information Matrices","volume":"49","author":"Stoica","year":"2001","journal-title":"IEEE Trans. Signal Proc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1049\/iet-rsn:20070041","article-title":"Fundamental resolution limits of closely spaced random signals","volume":"2","author":"Amari","year":"2008","journal-title":"IET Radar Sonar Navig."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"5309","DOI":"10.1109\/TSP.2008.929654","article-title":"Fundamental limitations on the resolution of deterministic signals","volume":"56","author":"Amari","year":"2008","journal-title":"IEEE Trans. Signal Proc."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Hero, A.O., Casta\u00f1\u00f3n, D., Cochran, D., and Kastella, K. (2008). Foundations and Applications of Sensor Management, Springer.","DOI":"10.1007\/978-0-387-49819-5"},{"key":"ref_22","first-page":"162","article-title":"Observability Criteria for Bearings-Only Target Motion Analysis","volume":"17","author":"Nardone","year":"1981","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1290","DOI":"10.1109\/7.543850","article-title":"Observability in Passive Target Motion Analysis","volume":"32","author":"Jauffret","year":"1996","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3084","DOI":"10.1109\/TAES.2012.6324679","article-title":"On Information Resolution of Radar Systems","volume":"48","author":"Cheng","year":"2012","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_25","unstructured":"Cheng, Y., Wang, X., and Moran, B. (2010, January 26\u201329). Sensor Network Performance Evaluation in Statistical Manifolds. Proceedings of the 2010 13th Conference on Information Fusion (FUSION), Edinburgh, UK."},{"key":"ref_26","unstructured":"Barton, D.K., and Leonov, S.A. (1997). Radar Technology Encyclopedia, Artech House."},{"key":"ref_27","unstructured":"Amari, S., and Nagaoka, H. (2000). Methods of Information Geometry, American Mathematical Society."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Lang, S. (1999). Foundations of Differential Geometry, Springer.","DOI":"10.1007\/978-1-4612-0541-8"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1214\/aos\/1176345779","article-title":"Differential Geometry of Curved Exponential Families-Curvatures and Information Loss","volume":"10","author":"Amari","year":"1982","journal-title":"Ann. Stat."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/3\/131\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:30:43Z","timestamp":1760207443000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/3\/131"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,17]]},"references-count":29,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,3]]}},"alternative-id":["e19030131"],"URL":"https:\/\/doi.org\/10.3390\/e19030131","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2017,3,17]]}}}