{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T19:09:32Z","timestamp":1775243372103,"version":"3.50.1"},"publisher-location":"Cham","reference-count":50,"publisher":"Springer International Publishing","isbn-type":[{"value":"9783030012335","type":"print"},{"value":"9783030012342","type":"electronic"}],"license":[{"start":{"date-parts":[[2018,1,1]],"date-time":"2018-01-01T00:00:00Z","timestamp":1514764800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2018,1,1]],"date-time":"2018-01-01T00:00:00Z","timestamp":1514764800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018]]},"DOI":"10.1007\/978-3-030-01234-2_37","type":"book-chapter","created":{"date-parts":[[2018,10,5]],"date-time":"2018-10-05T12:13:11Z","timestamp":1538741591000},"page":"621-637","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":33,"title":["Statistically-Motivated Second-Order Pooling"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0186-3399","authenticated-orcid":false,"given":"Kaicheng","family":"Yu","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8347-8637","authenticated-orcid":false,"given":"Mathieu","family":"Salzmann","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,10,6]]},"reference":[{"key":"37_CR1","unstructured":"Abadi, M., et al.: TensorFlow: large-scale machine learning on heterogeneous systems (2015). Software available from tensorflow.org. https:\/\/www.tensorflow.org\/"},{"key":"37_CR2","doi-asserted-by":"crossref","unstructured":"Arandjelovic, R., Zisserman, A.: All about VLAD. In: CVPR, pp. 1578\u20131585 (2013)","DOI":"10.1109\/CVPR.2013.207"},{"key":"37_CR3","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1002\/mrm.20965","volume":"56","author":"V Arsigny","year":"2006","unstructured":"Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn. Reson. Med. 56, 411\u2013421 (2006)","journal-title":"Magn. Reson. Med."},{"issue":"1","key":"37_CR4","doi-asserted-by":"publisher","first-page":"128","DOI":"10.2307\/2983618","volume":"8","author":"M. S. Bartlett","year":"1946","unstructured":"Bartlett, M.S., Kendall, D.G.: The statistical analysis of variance-heterogeneity and the logarithmic transformation. Suppl. J. R. Stat. Soc. 8(1), 128\u2013138 (1946). http:\/\/www.jstor.org\/stable\/2983618","journal-title":"Supplement to the Journal of the Royal Statistical Society"},{"key":"37_CR5","doi-asserted-by":"crossref","unstructured":"Bell, S., Upchurch, P., Snavely, N., Bala, K.: Material recognition in the wild with the materials in context database. In: CVPR (2015)","DOI":"10.1109\/CVPR.2015.7298970"},{"key":"37_CR6","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"430","DOI":"10.1007\/978-3-642-33786-4_32","volume-title":"Computer Vision \u2013 ECCV 2012","author":"J Carreira","year":"2012","unstructured":"Carreira, J., Caseiro, R., Batista, J., Sminchisescu, C.: Semantic segmentation with second-order pooling. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds.) ECCV 2012. LNCS, vol. 7578, pp. 430\u2013443. Springer, Heidelberg (2012). https:\/\/doi.org\/10.1007\/978-3-642-33786-4_32"},{"key":"37_CR7","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1007\/978-3-319-10578-9_20","volume-title":"Computer Vision \u2013 ECCV 2014","author":"A Cherian","year":"2014","unstructured":"Cherian, A., Sra, S.: Riemannian sparse coding for positive definite matrices. In: Fleet, D., Pajdla, T., Schiele, B., Tuytelaars, T. (eds.) ECCV 2014. LNCS, vol. 8691, pp. 299\u2013314. Springer, Cham (2014). https:\/\/doi.org\/10.1007\/978-3-319-10578-9_20"},{"key":"37_CR8","unstructured":"Chollet, F., et al.: Keras (2015). https:\/\/github.com\/fchollet\/keras"},{"key":"37_CR9","doi-asserted-by":"crossref","unstructured":"Cimpoi, M., Maji, S., Kokkinos, I., Mohamed, S., Vedaldi, A.: Describing textures in the wild. In: CVPR (2014)","DOI":"10.1109\/CVPR.2014.461"},{"key":"37_CR10","doi-asserted-by":"crossref","unstructured":"Cui, Y., Zhou, F., Wang, J., Liu, X., Lin, Y., Belongie, S.: Kernel pooling for convolutional neural networks. In: CVPR (2017)","DOI":"10.1109\/CVPR.2017.325"},{"key":"37_CR11","doi-asserted-by":"crossref","unstructured":"Dalal, N., Triggs, B.: Histograms of oriented gradients for human detection. In: CVPR, pp. 886\u2013893 (2005)","DOI":"10.1109\/CVPR.2005.177"},{"key":"37_CR12","first-page":"119","volume":"55","author":"Y Freund","year":"1997","unstructured":"Freund, Y., Schapire, R.E.: A decision-theoretic generalization of on-line learning and an application to boosting. Suppl. J. R. Stat. Soc. 55, 119\u2013139 (1997)","journal-title":"Suppl. J. R. Stat. Soc."},{"key":"37_CR13","doi-asserted-by":"crossref","unstructured":"Gao, Y., Beijbom, O., Zhang, N., Darrell, T.: Compact bilinear pooling. In: CVPR, pp. 317\u2013326 (2016)","DOI":"10.1109\/CVPR.2016.41"},{"key":"37_CR14","unstructured":"Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: AISTATS (2010)"},{"key":"37_CR15","unstructured":"Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press, Cambridge (2016). http:\/\/www.deeplearningbook.org"},{"key":"37_CR16","doi-asserted-by":"crossref","unstructured":"Guo, K., Ishwar, P., Konrad, J.: Action recognition using sparse representation on covariance manifolds of optical flow. In: IEEE International Conference on Advanced Video and Signal Based Surveillance (AVSS) (2010)","DOI":"10.1109\/AVSS.2010.71"},{"key":"37_CR17","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1007\/978-3-319-10605-2_2","volume-title":"Computer Vision \u2013 ECCV 2014","author":"MT Harandi","year":"2014","unstructured":"Harandi, M.T., Salzmann, M., Hartley, R.: From manifold to manifold: geometry-aware dimensionality reduction for SPD matrices. In: Fleet, D., Pajdla, T., Schiele, B., Tuytelaars, T. (eds.) ECCV 2014. LNCS, vol. 8690, pp. 17\u201332. Springer, Cham (2014). https:\/\/doi.org\/10.1007\/978-3-319-10605-2_2"},{"key":"37_CR18","doi-asserted-by":"crossref","unstructured":"Harandi, M., Salzmann, M.: Riemannian coding and dictionary learning: Kernels to the rescue. In: CVPR (2015)","DOI":"10.1109\/CVPR.2015.7299018"},{"key":"37_CR19","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"216","DOI":"10.1007\/978-3-642-33709-3_16","volume-title":"Computer Vision \u2013 ECCV 2012","author":"MT Harandi","year":"2012","unstructured":"Harandi, M.T., Sanderson, C., Hartley, R., Lovell, B.C.: Sparse coding and dictionary learning for symmetric positive definite matrices: a kernel approach. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds.) ECCV 2012. LNCS, pp. 216\u2013229. Springer, Heidelberg (2012). https:\/\/doi.org\/10.1007\/978-3-642-33709-3_16"},{"key":"37_CR20","doi-asserted-by":"crossref","unstructured":"He, K., Zhang, X., Ren, S., Sun, J.: Deep Residual Learning for Image Recognition. In: CVPR, pp. 770\u2013778 (2016)","DOI":"10.1109\/CVPR.2016.90"},{"key":"37_CR21","doi-asserted-by":"crossref","unstructured":"Huang, C.H., Boyer, E., Angonese, B.D.C., Navab, N., Ilic, S.: Toward user-specific tracking by detection of human shapes in multi-cameras. In: CVPR (2015)","DOI":"10.1109\/CVPR.2015.7299029"},{"key":"37_CR22","doi-asserted-by":"crossref","unstructured":"Huang, G., Liu, Z., Weinberger, K., van der Maaten, L.: Densely connected convolutional networks. In: CVPR (2017)","DOI":"10.1109\/CVPR.2017.243"},{"key":"37_CR23","unstructured":"Ioffe, S., Szegedy, C.: Batch normalization: accelerating deep network training by reducing internal covariate shift. In: ICML (2015)"},{"key":"37_CR24","doi-asserted-by":"crossref","unstructured":"Ionescu, C., Vantzos, O., Sminchisescu, C.: Matrix backpropagation for deep networks with structured layers (2015)","DOI":"10.1109\/ICCV.2015.339"},{"issue":"1178","key":"37_CR25","doi-asserted-by":"publisher","first-page":"364","DOI":"10.1098\/rspa.1955.0091","volume":"229","author":"A. T. James","year":"1955","unstructured":"James, A.T.: The non-central Wishart distribution. Proc. R. Soc. London. Ser. A Math. Phys. Sci. 229(1178), 364\u2013366 (1955). http:\/\/www.jstor.org\/stable\/99771","journal-title":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences"},{"key":"37_CR26","volume-title":"Applied Multivariate Statistical Analysis","author":"RA Johnson","year":"2014","unstructured":"Johnson, R.A., Wichern, D.W., et al.: Applied Multivariate Statistical Analysis, vol. 4. Prentice-Hall, Englewood Cliffs (2014)"},{"key":"37_CR27","doi-asserted-by":"crossref","unstructured":"Kong, S., Fowlkes, C.: Low-rank bilinear pooling for fine-grained classification. In: CVPR (2017)","DOI":"10.1109\/CVPR.2017.743"},{"key":"37_CR28","doi-asserted-by":"crossref","unstructured":"Koniusz, P., Tas, Y., Porikli, F.: Domain adaptation by mixture of alignments of second- or higher-order scatter tensors. In: CVPR (2017)","DOI":"10.1109\/CVPR.2017.755"},{"key":"37_CR29","doi-asserted-by":"crossref","unstructured":"Koniusz, P., Zhang, H., Porikli, F.: A deeper look at power normalizations. In: CVPR, pp. 5774\u20135783 (2018)","DOI":"10.1109\/CVPR.2018.00605"},{"key":"37_CR30","unstructured":"Krizhevsky, A., Sutskever, I., Hinton, G.: ImageNet classification with deep convolutional neural networks. In: NIPS, pp. 1106\u20131114 (2012)"},{"key":"37_CR31","unstructured":"Lazebnik, S., Schmid, C., Ponce, J.: Beyond bags of features: spatial pyramid matching for recognizing natural scene categories. In: CVPR (2006)"},{"key":"37_CR32","doi-asserted-by":"crossref","unstructured":"Li, P., Xie, J., Wang, Q., Zuo, W.: Is second-order information helpful for large-scale visual recognition? In: ICCV (2017)","DOI":"10.1109\/ICCV.2017.228"},{"key":"37_CR33","doi-asserted-by":"crossref","unstructured":"Li, P., Wang, Q., Zuo, W., Zhang, L.: Log-Euclidean kernels for sparse representation and dictionary learning. In: ICCV (2013)","DOI":"10.1109\/ICCV.2013.202"},{"key":"37_CR34","doi-asserted-by":"crossref","unstructured":"Lin, T.Y., Maji, S.: Improved bilinear pooling with CNNs. In: BMVC (2017)","DOI":"10.5244\/C.31.117"},{"key":"37_CR35","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1007\/978-3-030-01219-9_38","volume-title":"Computer Vision \u2013 ECCV 2018","author":"TY Lin","year":"2018","unstructured":"Lin, T.Y., Maji, S., Koniusz, P.: Second-order democratic aggregation. In: Ferrari, V., Hebert, M., Sminchisescu, C., Weiss, Y. (eds.) ECCV 2018, Part III. LNCS, vol. 11207, pp. 639\u2013656. Springer, Cham (2018)"},{"key":"37_CR36","doi-asserted-by":"crossref","unstructured":"Lin, T., RoyChowdhury, A., Maji, S.: Bilinear CNN models for fine-grained visual recognition. In: ICCV, pp. 1449\u20131457 (2015)","DOI":"10.1109\/ICCV.2015.170"},{"key":"37_CR37","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.: A Riemannian framework for tensor computing. IJCV 66, 41\u201366 (2006)","journal-title":"IJCV"},{"key":"37_CR38","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1007\/978-3-642-15561-1_11","volume-title":"Computer Vision \u2013 ECCV 2010","author":"F Perronnin","year":"2010","unstructured":"Perronnin, F., S\u00e1nchez, J., Mensink, T.: Improving the Fisher kernel for large-scale image classification. In: Daniilidis, K., Maragos, P., Paragios, N. (eds.) ECCV 2010. LNCS, vol. 6314, pp. 143\u2013156. Springer, Heidelberg (2010). https:\/\/doi.org\/10.1007\/978-3-642-15561-1_11"},{"key":"37_CR39","unstructured":"Quang, M.H., San-Biagio, M., Murino, V.: Log-Hilbert-Schmidt metric between positive definite operators on Hilbert spaces. In: NIPS (2014)"},{"key":"37_CR40","doi-asserted-by":"crossref","unstructured":"Quattoni, A., Torralba, A.: Recognizing indoor scenes. In: CVPR, pp. 413\u2013420 (2009)","DOI":"10.1109\/CVPR.2009.5206537"},{"key":"37_CR41","unstructured":"Sermanet, P., Chintala, S., LeCun, Y.: Convolutional neural networks applied to house numbers digit classification. In: ICPR (2012)"},{"key":"37_CR42","unstructured":"Simonyan, K., Zisserman, A.: Very deep convolutional networks for large-scale image recognition. In: ICLR (2015)"},{"key":"37_CR43","unstructured":"Sra, S.: A new metric on the manifold of kernel matrices with application to matrix geometric means. In: NIPS (2012)"},{"key":"37_CR44","series-title":"Lecture Notes in computer science (lecture notes in artificial intelligence)","doi-asserted-by":"publisher","first-page":"318","DOI":"10.1007\/978-3-642-23808-6_21","volume-title":"Machine Learning and Knowledge Discovery in Databases","author":"S Sra","year":"2011","unstructured":"Sra, S., Cherian, A.: Generalized dictionary learning for symmetric positive definite matrices with application to nearest neighbor retrieval. In: Gunopulos, D., Hofmann, T., Malerba, D., Vazirgiannis, M. (eds.) ECML PKDD 2011. LNCS (LNAI), vol. 6913, pp. 318\u2013332. Springer, Heidelberg (2011). https:\/\/doi.org\/10.1007\/978-3-642-23808-6_21"},{"key":"37_CR45","doi-asserted-by":"crossref","unstructured":"Szegedy, C., et al.: Going deeper with convolutions. In: CVPR, pp. 1\u20139, June 2015","DOI":"10.1109\/CVPR.2015.7298594"},{"key":"37_CR46","doi-asserted-by":"crossref","unstructured":"Tuzel, O., Porikli, F., Meer, P.: Human detection via classification on Riemannian manifolds. In: CVPR, pp. 1\u20138 (2007)","DOI":"10.1109\/CVPR.2007.383197"},{"key":"37_CR47","unstructured":"Vapnik, V.: Statistical Learning Theory. Wiley-Interscience, New York (1998)"},{"key":"37_CR48","unstructured":"Wah, C., Branson, S., Welinder, P., Perona, P., Belongie, S.: The Caltech-UCSD birds-200-2011 dataset. Technical report (2011)"},{"key":"37_CR49","doi-asserted-by":"crossref","unstructured":"Wang, Q., Li, P., Zuo, W., Zhang, L.: RAID-G - robust estimation of approximate infinite dimensional Gaussian with application to material recognition. In: CVPR (2016)","DOI":"10.1109\/CVPR.2016.480"},{"issue":"12","key":"37_CR50","doi-asserted-by":"publisher","first-page":"684","DOI":"10.1073\/pnas.17.12.684","volume":"17","author":"EB Wilson","year":"1931","unstructured":"Wilson, E.B., Hilferty, M.M.: The distribution of chi-square. Proc. Natl. Acad. Sci. 17(12), 684\u2013688 (1931)","journal-title":"Proc. Natl. Acad. Sci."}],"container-title":["Lecture Notes in Computer Science","Computer Vision \u2013 ECCV 2018"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-030-01234-2_37","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T18:33:59Z","timestamp":1775241239000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-030-01234-2_37"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018]]},"ISBN":["9783030012335","9783030012342"],"references-count":50,"URL":"https:\/\/doi.org\/10.1007\/978-3-030-01234-2_37","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018]]},"assertion":[{"value":"6 October 2018","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"ECCV","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"European Conference on Computer Vision","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Munich","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Germany","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2018","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"8 September 2018","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"14 September 2018","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"15","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"eccv2018","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/eccv2018.org\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}