{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:21:38Z","timestamp":1740108098998,"version":"3.37.3"},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2021,1,2]],"date-time":"2021-01-02T00:00:00Z","timestamp":1609545600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,1,2]],"date-time":"2021-01-02T00:00:00Z","timestamp":1609545600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61773022"],"award-info":[{"award-number":["61773022"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Pattern Anal Applic"],"published-print":{"date-parts":[[2021,5]]},"DOI":"10.1007\/s10044-020-00941-1","type":"journal-article","created":{"date-parts":[[2021,1,2]],"date-time":"2021-01-02T02:02:39Z","timestamp":1609552959000},"page":"723-739","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The framework of learnable kernel function and its application to dictionary learning of SPD data"],"prefix":"10.1007","volume":"24","author":[{"given":"Weijia","family":"Feng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6553-1070","authenticated-orcid":false,"given":"Zhengming","family":"Ma","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rixin","family":"Zhuang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hangjian","family":"Che","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,1,2]]},"reference":[{"key":"941_CR1","doi-asserted-by":"crossref","unstructured":"Turk M, Pentland AP (1991) Face recognition using eigenfaces. In: IEEE conference on computer vision and pattern recognition, pp 586\u2013591","DOI":"10.1109\/CVPR.1991.139758"},{"issue":"7","key":"941_CR2","doi-asserted-by":"publisher","first-page":"711","DOI":"10.1109\/34.598228","volume":"19","author":"PN Belhumeur","year":"1997","unstructured":"Belhumeur PN, Hespanha JP, Kriengman DJ (1997) Eigenfaces vs. Fisherfaces: recognition using class specific linear projection. IEEE Trans Pattern Anal Mach Intell 19(7):711\u2013720","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"key":"941_CR3","doi-asserted-by":"crossref","unstructured":"Wan M, Li M, Yang G et al (2014) Feature extraction using two-dimensional maximum embedding difference. In: Information sciences, pp 55\u201369","DOI":"10.1016\/j.ins.2014.02.145"},{"key":"941_CR4","doi-asserted-by":"crossref","unstructured":"Wan M, Yang G, Gai S et al (2017) Two-dimensional discriminant locality preserving projections (2DDLPP) and its application to feature extraction via fuzzy set. In: Multimed tools appl, vol 76, pp 355\u2013371. https:\/\/doi.org\/10.1007\/s11042-015-3057-8","DOI":"10.1007\/s11042-015-3057-8"},{"key":"941_CR5","doi-asserted-by":"publisher","first-page":"107456","DOI":"10.1016\/j.sigpro.2020.107456","volume":"170","author":"Z Liu","year":"2020","unstructured":"Liu Z, Lai Z, Ou W et al (2020) Structured optimal graph based sparse feature extraction for semi-supervised learning. Sig Process 170:107456","journal-title":"Sig Process"},{"issue":"1","key":"941_CR6","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1007\/s11263-005-3222-z","volume":"66","author":"X Pennec","year":"2006","unstructured":"Pennec X, Fillard P, Ayache N (2006) A Riemannian framework for tensor computing. Int J Comput Vis 66(1):41\u201366","journal-title":"Int J Comput Vis"},{"issue":"4","key":"941_CR7","doi-asserted-by":"publisher","first-page":"2153","DOI":"10.1109\/TIT.2017.2653803","volume":"63","author":"S Said","year":"2017","unstructured":"Said S, Bombrun L, Berthoumieu Y, Manton JH (2017) Riemannian Gaussian distributions on the space of symmetric positive definite matrices. IEEE Trans Inform Theory 63(4):2153\u20132170","journal-title":"IEEE Trans Inform Theory"},{"key":"941_CR8","unstructured":"Wang R, Guo H, Davis LS et al (2012) Covariance discriminative learning: a natural and efficient approach to image set classification. In: IEEE conference on computer vision and pattern recognition, pp 2496\u20132503"},{"issue":"6","key":"941_CR9","doi-asserted-by":"publisher","first-page":"2479","DOI":"10.1109\/TIP.2013.2252622","volume":"22","author":"K Guo","year":"2013","unstructured":"Guo K, Ishwar P, Konrad J (2013) Action recognition from video using feature covariance matrices. IEEE Trans Image Process 22(6):2479\u20132494","journal-title":"IEEE Trans Image Process"},{"issue":"10","key":"941_CR10","doi-asserted-by":"publisher","first-page":"1713","DOI":"10.1109\/TPAMI.2008.75","volume":"30","author":"O Tuzel","year":"2008","unstructured":"Tuzel O, Porikli F, Meer P (2008) Pedestrian detection via classification on Riemannian manifolds. IEEE Trans Pattern Anal Mach Intell 30(10):1713\u20131727. https:\/\/doi.org\/10.1109\/TPAMI.2008.75","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"issue":"2","key":"941_CR11","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1007\/s11263-012-0582-z","volume":"101","author":"T Zhang","year":"2013","unstructured":"Zhang T, Ghanem B, Liu S et al (2013) Robust visual tracking via structured multi-task sparse learning. Int J Comput Vis 101(2):367\u2013383","journal-title":"Int J Comput Vis"},{"key":"941_CR12","doi-asserted-by":"crossref","unstructured":"Yang M, Zhang L (2010) Gabor feature based sparse representation for face recognition with Gabor occlusion dictionary. In: ECCV, pp 448\u2013461","DOI":"10.1007\/978-3-642-15567-3_33"},{"key":"941_CR13","doi-asserted-by":"crossref","unstructured":"Wan M et al (2016) Local graph embedding based on maximum margin criterion via fuzzy set. In: Fuzzy sets and systems, vol 318.JUL.1, pp 120\u2013131","DOI":"10.1016\/j.fss.2016.06.001"},{"key":"941_CR14","doi-asserted-by":"crossref","unstructured":"Liu Z, Wang J, Liu G et al (2013) Discriminative low-rank preserving projection for dimensionality reduction. In: Applied soft computing, vol 85, pp 105768, 2019. J. S. Turner, Log-Euclidean kernels for sparse representation and dictionary learning, ICCV, pp 1601\u20131608","DOI":"10.1016\/j.asoc.2019.105768"},{"key":"941_CR15","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-7011-4","volume-title":"Sparse and redundant representations\u2013from theory to applications in signal and image processing","author":"M Elad","year":"2010","unstructured":"Elad M (2010) Sparse and redundant representations\u2013from theory to applications in signal and image processing. Springer Publishing Company, Berlin"},{"key":"941_CR16","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511809682","volume-title":"Kernel methods for pattern analysis","author":"J Shawetaylor","year":"2004","unstructured":"Shawetaylor J, Cristianini N (2004) Kernel methods for pattern analysis. Cambridge University Press, New York"},{"key":"941_CR17","doi-asserted-by":"crossref","unstructured":"Harandi M, Sanderson C, Hartley R, Lovell B (2012) Sparse coding and dictionary learning for symmetric positive definite matrices: a kernel approach. In: ECCV, pp 216\u2013229","DOI":"10.1007\/978-3-642-33709-3_16"},{"key":"941_CR18","doi-asserted-by":"crossref","unstructured":"Jayasumana S, Hartley R, Salzmann M, Li H, Harandi M (2013) Kernel methods on the Riemannian manifold of symmetric positive definite matrices. In: CVPR, pp 73\u201380","DOI":"10.1109\/CVPR.2013.17"},{"key":"941_CR19","doi-asserted-by":"crossref","unstructured":"Vemulapalli R, Pillai JK, Chellappa R (2013) Kernel learning for extrinsic classification of manifold features. In: IEEE conference on computer vision and pattern recognition (CVPR)","DOI":"10.1109\/CVPR.2013.233"},{"issue":"4","key":"941_CR20","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1109\/34.761261","volume":"21","author":"T Randen","year":"1999","unstructured":"Randen T, Husoy JH (1999) Filtering for texture classification: a comparative study. IEEE Trans Pattern Anal Mach Intell 21(4):291\u2013310","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"key":"941_CR21","doi-asserted-by":"crossref","unstructured":"Goh A, Vidal R (2008) Clustering and dimensionality reduction on Riemannian manifolds. In: IEEE conference on computer vision and pattern recognition, pp 1\u20137","DOI":"10.1109\/CVPR.2008.4587422"},{"key":"941_CR22","doi-asserted-by":"crossref","unstructured":"Harandi M, Sanderson C, Wiliem A et al (2012) Kernel analysis over Riemannian manifolds for visual recognition of actions, pedestrians and textures","DOI":"10.1109\/WACV.2012.6163005"},{"key":"941_CR23","doi-asserted-by":"crossref","unstructured":"Jayasumana S, Hartley R, Salzmann M et al (2015) Kernel methods on riemannian manifolds with Gaussian RBF Kernels. In: IEEE transactions on pattern analysis & machine intelligence, vol 37, no 12, pp 1\u20131","DOI":"10.1109\/TPAMI.2015.2414422"},{"issue":"11","key":"941_CR24","doi-asserted-by":"publisher","first-page":"3729","DOI":"10.1109\/TIP.2015.2451953","volume":"24","author":"Y Wu","year":"2015","unstructured":"Wu Y, Jia Y, Li P, Zhang J, Yuan J (2015) Manifold kernel sparse representation of symmetric positive-definite matrices and its applications. IEEE Trans Image Process 24(11):3729\u20133741","journal-title":"IEEE Trans Image Process"},{"issue":"6","key":"941_CR25","doi-asserted-by":"publisher","first-page":"1294","DOI":"10.1109\/TNNLS.2014.2387383","volume":"27","author":"M Harandi","year":"2014","unstructured":"Harandi M, Hartley R, Lovell B et al (2014) Sparse coding on symmetric positive definite manifolds using bregman divergences. IEEE Trans Neural Netw Learn Syst 27(6):1294\u20131306","journal-title":"IEEE Trans Neural Netw Learn Syst"},{"issue":"3","key":"941_CR26","doi-asserted-by":"publisher","first-page":"592","DOI":"10.1109\/TPAMI.2013.143","volume":"36","author":"R Sivalingam","year":"2014","unstructured":"Sivalingam R, Boley D, Morellas V et al (2014) Tensor sparse coding for positive definite matrices. IEEE Trans Pattern Anal Mach Intell 36(3):592\u2013605","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"issue":"11","key":"941_CR27","doi-asserted-by":"publisher","first-page":"4592","DOI":"10.1109\/TIP.2015.2440766","volume":"24","author":"R Sivalingam","year":"2015","unstructured":"Sivalingam R, Boley D, Morellas V et al (2015) Tensor dictionary learning for positive definite matrices. IEEE Trans Image Process 24(11):4592","journal-title":"IEEE Trans Image Process"},{"issue":"11","key":"941_CR28","first-page":"2859","volume":"28","author":"A Cherian","year":"2016","unstructured":"Cherian A, Sra S (2016) Riemannian dictionary learning and sparse coding for positive definite matrices. IEEE Trans Neural Netw Learn Syst 28(11):2859\u20132871","journal-title":"IEEE Trans Neural Netw Learn Syst"},{"issue":"7","key":"941_CR29","doi-asserted-by":"publisher","first-page":"1750","DOI":"10.1016\/j.patcog.2012.08.015","volume":"46","author":"X Zhang","year":"2013","unstructured":"Zhang X, Li W, Hu W et al (2013) Block covariance based L1 tracker with a subtle template dictionary. Pattern Recogn 46(7):1750\u20131761","journal-title":"Pattern Recogn"},{"key":"941_CR30","doi-asserted-by":"crossref","unstructured":"Guo K, Ishwar P, Konrad J (2010) Action recognition using sparse representation on covariance manifolds of optical flow. In: IEEE international conference on advanced video and signal based surveillance, pp 188\u2013195","DOI":"10.1109\/AVSS.2010.71"},{"key":"941_CR31","doi-asserted-by":"crossref","unstructured":"Yuan C, Hu W, Li X et al (2010) Human action recognition under log-euclidean riemannian metric. In: Asian conference on computer vision, pp 343\u2013353","DOI":"10.1007\/978-3-642-12307-8_32"},{"key":"941_CR32","doi-asserted-by":"crossref","unstructured":"Li P, Wang Q, Zuo W et al (2013) Log-Euclidean kernels for sparse representation and dictionary learning. In: ICCV, pp 1601\u20131608","DOI":"10.1109\/ICCV.2013.202"},{"issue":"3","key":"941_CR33","doi-asserted-by":"publisher","first-page":"1248","DOI":"10.1109\/TIP.2018.2877337","volume":"28","author":"A Das","year":"2019","unstructured":"Das A, Nair MS, Peter SD (2019) Sparse representation over learned dictionaries on the Riemannian manifold for automated grading of nuclear pleomorphism in breast cancer. IEEE Trans Image Process 28(3):1248\u20131260. https:\/\/doi.org\/10.1109\/TIP.2018.2877337","journal-title":"IEEE Trans Image Process"},{"key":"941_CR34","doi-asserted-by":"publisher","unstructured":"Li D, Chen L, Wang F (2016) Semantic and neighborhood preserving dictionary learning for symmetric positive-definite matrices. In: IEEE 13th international conference on signal processing (ICSP), Chengdu, pp 654\u2013658. https:\/\/doi.org\/10.1109\/ICSP.2016.7877913","DOI":"10.1109\/ICSP.2016.7877913"},{"key":"941_CR35","unstructured":"[Online]. Available: http:\/\/spams-devel.gforge.inria.fr\/"},{"key":"941_CR36","unstructured":"[Online]. Available: http:\/\/cvxr.com\/cvx\/"},{"issue":"3","key":"941_CR37","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/s10208-005-0179-9","volume":"7","author":"PA Absil","year":"2007","unstructured":"Absil PA, Baker CG, Gallivan KA (2007) Trust-region methods on Riemannian manifolds. Found Comput Math 7(3):303\u2013330","journal-title":"Found Comput Math"},{"key":"941_CR38","doi-asserted-by":"publisher","DOI":"10.1515\/9781400830244","volume-title":"Optimization algorithms on matrix manifolds","author":"PA Absil","year":"2008","unstructured":"Absil PA, Mahony R, Sepulchre R (2008) Optimization algorithms on matrix manifolds. Princeton University Press, Princeton"},{"key":"941_CR39","unstructured":"[Online]. Available: https:\/\/www.manopt.org\/"},{"issue":"8","key":"941_CR40","doi-asserted-by":"publisher","first-page":"1972","DOI":"10.1109\/TPAMI.2012.263","volume":"35","author":"D Tosato","year":"2013","unstructured":"Tosato D, Spera M et al (2013) Characterizing humans on Riemannian manifolds. IEEE Trans Pattern Anal Mach Intell 35(8):1972\u20131984","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"key":"941_CR41","doi-asserted-by":"crossref","unstructured":"Sra S, Cherian A (2011) Generalized dictionary learning for symmetric positive definite matrices with application to nearest neighbor retrieval. In: Proceedings of European conference on machine learning, pp 318\u2013332","DOI":"10.1007\/978-3-642-23808-6_21"},{"issue":"10","key":"941_CR42","doi-asserted-by":"publisher","first-page":"1090","DOI":"10.1109\/34.879790","volume":"22","author":"PJ Phillips","year":"2000","unstructured":"Phillips PJ, Moon H, Rizvi SA et al (2000) The FERET evaluation methodology for face-recognition algorithms. IEEE Trans Pattern Anal Mach Intell 22(10):1090\u20131104","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"issue":"6","key":"941_CR43","doi-asserted-by":"publisher","first-page":"2479","DOI":"10.1109\/TIP.2013.2252622","volume":"22","author":"K Guo","year":"2013","unstructured":"Guo K, Ishwar P, Konrad J (2013) Action recognition from video using feature covariance matrices. IEEE Trans Image Process 22(6):2479\u20132494","journal-title":"IEEE Trans Image Process"},{"issue":"2","key":"941_CR44","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1109\/TPAMI.2008.79","volume":"32","author":"J Wright","year":"2009","unstructured":"Wright J, Yang A, Ganesh A, Sastry S, Ma Y (2009) Robust face recognition via sparse representation. IEEE Trans Pattern Anal Mach Intell 32(2):210\u2013227","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"key":"941_CR45","doi-asserted-by":"crossref","unstructured":"Yang M, Zhang L (2010) Gabor feature based sparse representation for face recognition with Gabor occlusion dictionary. In: ECCV, pp 448\u2013461","DOI":"10.1007\/978-3-642-15567-3_33"},{"key":"941_CR46","doi-asserted-by":"crossref","unstructured":"Schwartz WR, Davis LS (2009) Learning discriminative appearance based models using partial least squares. In: Brazilian Symposium on Computer Graphics and Image Processing. Brazil, Oct, Rio de Janeiro, pp 322\u2013329","DOI":"10.1109\/SIBGRAPI.2009.42"},{"issue":"5","key":"941_CR47","doi-asserted-by":"publisher","first-page":"687","DOI":"10.1109\/TNN.2011.2109395","volume":"22","author":"PA Huy","year":"2011","unstructured":"Huy PA, Andrzej C (2011) Extended Hamiltonian learning on Riemannian manifolds: theoretical aspects. IEEE Trans Neural Netw 22(5):687\u2013700","journal-title":"IEEE Trans Neural Netw"}],"container-title":["Pattern Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10044-020-00941-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10044-020-00941-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10044-020-00941-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,19]],"date-time":"2021-10-19T03:37:39Z","timestamp":1634614659000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10044-020-00941-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,2]]},"references-count":47,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,5]]}},"alternative-id":["941"],"URL":"https:\/\/doi.org\/10.1007\/s10044-020-00941-1","relation":{},"ISSN":["1433-7541","1433-755X"],"issn-type":[{"type":"print","value":"1433-7541"},{"type":"electronic","value":"1433-755X"}],"subject":[],"published":{"date-parts":[[2021,1,2]]},"assertion":[{"value":"30 January 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 November 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 January 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}