{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:34:48Z","timestamp":1787333688912,"version":"build-2736575974"},"reference-count":41,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>We present a new Riemannian metric, termed Log-Cholesky metric, on the manifold of symmetric positive definite (SPD) matrices via Cholesky decomposition. We first construct a Lie group structure and a bi-invariant metric on Cholesky space, the collection of lower triangular matrices whose diagonal elements are all positive. Such group structure and metric are then pushed forward to the space of SPD matrices via the inverse of Cholesky decomposition that is a bijective map between Cholesky space and SPD matrix space. This new Riemannian metric and Lie group structure fully circumvent swelling effect, in the sense that the determinant of the Fr\u00e9chet average of a set of SPD matrices under the presented metric, called Log-Cholesky average, is between the minimum and the maximum of the determinants of the original SPD matrices. Comparing to existing metrics such as the affine-invariant metric and Log-Euclidean metric, the presented metric is simpler, more computationally efficient, and numerically stabler. In particular, parallel transport along geodesics under Log-Cholesky metric is given in a closed and easy-to-compute form.<\/jats:p>","DOI":"10.1137\/18m1221084","type":"journal-article","created":{"date-parts":[[2019,11,20]],"date-time":"2019-11-20T09:44:40Z","timestamp":1574243080000},"page":"1353-1370","source":"Crossref","is-referenced-by-count":93,"title":["Riemannian Geometry of Symmetric Positive Definite Matrices via Cholesky Decomposition"],"prefix":"10.1137","volume":"40","author":[{"given":"Zhenhua","family":"Lin","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,11,19]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"K. Aftab and R. Hartley,\n                      Lq averaging for symmetric positive-definite matrices\n                      , in Proceedings of the International Conference on Digital Image Computing: Techniques and Applications (DICTA), 2013.","DOI":"10.1109\/DICTA.2013.6691505"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2003.11.019"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1109\/JSTSP.2013.2261798"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/050637996"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1109\/TBME.2011.2172210"},{"key":"atypb6","unstructured":"R. Bhatia,\n                      Positive Definite Matrices\n                      , Princeton University Press, 2007."},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/j.exmath.2018.01.002"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1046294456"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/j.patcog.2012.04.011"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2011.12.003"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1214\/09-AOAS249"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1198\/000313002753631349"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuroimage.2006.09.027"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/j.sigpro.2005.12.018"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1089\/brain.2011.0008"},{"key":"atypb16","unstructured":"G. H. Golub and C. F. Van Loan,\n                      Matrix Computations\n                      , 3rd ed., Johns Hopkins University Press, 1996."},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2006.11.024"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1002\/nme.3338"},{"key":"atypb19","doi-asserted-by":"crossref","unstructured":"S. Helgason,\n                      Differential Geometry, Lie Groups, and Symmetric Spaces\n                      , Amer. Math. Soc., 2001.","DOI":"10.1090\/gsm\/034"},{"key":"atypb20","unstructured":"R. Hosseini and S. Sra,\n                      Matrix manifold optimization for Gaussian mixtures\n                      , in Proceedings of the NIPS, 2015."},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/j.dsp.2017.06.019"},{"key":"atypb22","unstructured":"S. A. Huettel, A. W. Song, and G. McCarthy,\n                      Functional Magnetic Resonance Imaging\n                      , 2nd ed., Sinauer Associates, 2008."},{"key":"atypb23","unstructured":"H. J. Kim, J. X. B. C. Vemuri, and V. Singh,\n                      Manifold-valued Dirichlet processes\n                      , in Proceedings of the 32nd International Conference on Machine Learning, vol. 37, 2015."},{"key":"atypb24","doi-asserted-by":"crossref","unstructured":"S. Lang,\n                      Differential and Riemannian Manifolds\n                      , Springer, New York, 1995.","DOI":"10.1007\/978-1-4612-4182-9"},{"key":"atypb25","first-page":"1","volume":"7","author":"Bihan D. Le","year":"1991","journal-title":"Magnetic Resonance Quarterly"},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"J. M. Lee,\n                      Manifolds and Differential Geometry\n                      , Graduate Studies in Mathematics, Amer. Math. Soc., 2009.","DOI":"10.1090\/gsm\/107"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-006-6897-z"},{"key":"atypb28","unstructured":"E. Massart and P.A. Absil,\n                      Quotient geometry with simple geodesics for the manifold of fixed-rank positive-semidefinite matrices\n                      , tech. report, ICTEAM Institute, UCLouvain, 2018."},{"key":"atypb29","unstructured":"J. Milnor,\n                      Morse Theory\n                      , Princeton University Press, 1963."},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479803436937"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1007\/s10659-005-9035-z"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmva.2013.04.006"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-006-6228-4"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1007\/s11263-005-3222-z"},{"key":"atypb35","doi-asserted-by":"crossref","unstructured":"Y. Rathi, A. Tannenbaum, and O. Michailovich,\n                      Segmenting images on the tensor manifold\n                      , in Proocedings of Computer Vision and Pattern Recognition, 2007.","DOI":"10.1109\/CVPR.2007.383010"},{"key":"atypb36","first-page":"1","volume":"18","author":"Schiratti J.-B.","year":"2017","journal-title":"J. Mach. Learn. Res."},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1090\/proc\/12953"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/drs006"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2004.831218"},{"key":"atypb40","doi-asserted-by":"crossref","unstructured":"Y. Yuan, H. Zhu, W. Lin, and J. S. Marron,\n                      Local polynomial regression for symmetric positive definite matrices\n                      , J. Royal Statistical Society: Series B (Statistical Methodology), 74 (2012), pp. 697-719.","DOI":"10.1111\/j.1467-9868.2011.01022.x"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1109\/LRA.2017.2657001"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1221084","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:17:58Z","timestamp":1787332678000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1221084"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":41,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M1221084"],"URL":"https:\/\/doi.org\/10.1137\/18m1221084","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}