{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,11]],"date-time":"2026-02-11T15:19:28Z","timestamp":1770823168348,"version":"3.50.1"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2018,6,27]],"date-time":"2018-06-27T00:00:00Z","timestamp":1530057600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"crossref","award":["61673201"],"award-info":[{"award-number":["61673201"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Mach Learn"],"published-print":{"date-parts":[[2019,2]]},"DOI":"10.1007\/s10994-018-5734-0","type":"journal-article","created":{"date-parts":[[2018,6,27]],"date-time":"2018-06-27T12:05:37Z","timestamp":1530101137000},"page":"267-295","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["Fast generalization rates for distance metric learning"],"prefix":"10.1007","volume":"108","author":[{"given":"Han-Jia","family":"Ye","sequence":"first","affiliation":[]},{"given":"De-Chuan","family":"Zhan","sequence":"additional","affiliation":[]},{"given":"Yuan","family":"Jiang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,6,27]]},"reference":[{"issue":"Feb","key":"5734_CR1","first-page":"441","volume":"10","author":"S Agarwal","year":"2009","unstructured":"Agarwal, S., & Niyogi, P. (2009). Generalization bounds for ranking algorithms via algorithmic stability. Journal of Machine Learning Research, 10(Feb), 441\u2013474.","journal-title":"Journal of Machine Learning Research"},{"issue":"4","key":"5734_CR2","doi-asserted-by":"publisher","first-page":"1497","DOI":"10.1214\/009053605000000282","volume":"33","author":"PL Bartlett","year":"2005","unstructured":"Bartlett, P. L., Bousquet, O., & Mendelson, S. (2005). Local rademacher complexities. The Annals of Statistics, 33(4), 1497\u20131537.","journal-title":"The Annals of Statistics"},{"issue":"Nov","key":"5734_CR3","first-page":"463","volume":"3","author":"PL Bartlett","year":"2002","unstructured":"Bartlett, P. L., & Mendelson, S. (2002). Rademacher and gaussian complexities: Risk bounds and structural results. Journal of Machine Learning Research, 3(Nov), 463\u2013482.","journal-title":"Journal of Machine Learning Research"},{"issue":"1","key":"5734_CR4","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1137\/080716542","volume":"2","author":"A Beck","year":"2009","unstructured":"Beck, A., & Teboulle, M. (2009). A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences, 2(1), 183\u2013202.","journal-title":"SIAM Journal on Imaging Sciences"},{"key":"5734_CR5","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1016\/j.neucom.2014.09.044","volume":"151","author":"A Bellet","year":"2015","unstructured":"Bellet, A., & Habrard, A. (2015). Robustness and generalization for metric learning. Neurocomputing, 151, 259\u2013267.","journal-title":"Neurocomputing"},{"key":"5734_CR6","unstructured":"Bellet, A., Habrard, A., & Sebban, M. (2012). Similarity learning for provably accurate sparse linear classification. In Proceedings of the 29th international conference on machine learning, Edinburgh, Scotland (pp. 1871\u20131878)."},{"key":"5734_CR7","volume-title":"Metric learning. Synthesis lectures on artificial intelligence and machine learning","author":"A Bellet","year":"2015","unstructured":"Bellet, A., Habrard, A., & Sebban, M. (2015). Metric learning. Synthesis lectures on artificial intelligence and machine learning. Rafael: Morgan & Claypool Publishers."},{"key":"5734_CR8","unstructured":"Bian, W., & Tao, D. (2011). Learning a distance metric by empirical loss minimization. In Proceedings of the 22nd international joint conference on artificial intelligence Barcelona, Spain (pp. 1186\u20131191)."},{"issue":"Mar","key":"5734_CR9","first-page":"499","volume":"2","author":"O Bousquet","year":"2002","unstructured":"Bousquet, O., & Elisseeff, A. (2002). Stability and generalization. Journal of Machine Learning Research, 2(Mar), 499\u2013526.","journal-title":"Journal of Machine Learning Research"},{"key":"5734_CR10","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511804441","volume-title":"Convex optimization","author":"S Boyd","year":"2004","unstructured":"Boyd, S., & Vandenberghe, L. (2004). Convex optimization. Cambridge: Cambridge University Press."},{"issue":"1","key":"5734_CR11","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1007\/s10994-015-5499-7","volume":"102","author":"Q Cao","year":"2016","unstructured":"Cao, Q., Guo, Z., & Ying, Y. (2016). Generalization bounds for metric and similarity learning. Machine Learning, 102(1), 115\u2013132.","journal-title":"Machine Learning"},{"key":"5734_CR12","first-page":"1511","volume-title":"Advances in neural information processing systems","author":"S Changpinyo","year":"2013","unstructured":"Changpinyo, S., Liu, K., & Sha, F. (2013). Similarity component analysis. Advances in neural information processing systems (Vol. 26, pp. 1511\u20131519). Cambridge: MIT Press."},{"key":"5734_CR13","first-page":"1109","volume":"11","author":"G Chechik","year":"2010","unstructured":"Chechik, G., Sharma, V., Shalit, U., & Bengio, S. (2010). Large scale online learning of image similarity through ranking. Journal of Machine Learning Research, 11, 1109\u20131135.","journal-title":"Journal of Machine Learning Research"},{"issue":"2","key":"5734_CR14","doi-asserted-by":"publisher","first-page":"844","DOI":"10.1214\/009052607000000910","volume":"36","author":"S Cl\u00e9men\u00e7on","year":"2008","unstructured":"Cl\u00e9men\u00e7on, S., Lugosi, G., & Vayatis, N. (2008). Ranking and empirical minimization of u-statistics. The Annals of Statistics, 36(2), 844\u2013874.","journal-title":"The Annals of Statistics"},{"key":"5734_CR15","doi-asserted-by":"crossref","unstructured":"Davis, J. V., Kulis, B., Jain, P., Sra, S., & Dhillon, I. S. (2007). Information-theoretic metric learning. In Proceedings of the 24th international conference on machine learning, Corvalis, OR (pp. 209\u2013216).","DOI":"10.1145\/1273496.1273523"},{"key":"5734_CR16","unstructured":"Do, H., Kalousis, A., Wang, J., & Woznica, A. (2012). A metric learning perspective of SVM: on the relation of LMNN and SVM. In Proceedings of the 15th international conference on artificial intelligence and statistics, April 21\u201323, 2012, La Palma, Canary Islands (pp 308\u2013317)."},{"key":"5734_CR17","first-page":"417","volume-title":"Advances in neural information processing systems","author":"A Frome","year":"2007","unstructured":"Frome, A., Singer, Y., & Malik, J. (2007). Image retrieval and classification using local distance functions. Advances in neural information processing systems (Vol. 19, pp. 417\u2013424). Cambridge, MA: MIT Press."},{"issue":"3","key":"5734_CR18","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1162\/NECO_a_00556","volume":"26","author":"Z Guo","year":"2014","unstructured":"Guo, Z., & Ying, Y. (2014). Guaranteed classification via regularized similarity learning. Neural Computation, 26(3), 497\u2013522.","journal-title":"Neural Computation"},{"key":"5734_CR19","doi-asserted-by":"crossref","unstructured":"Hsieh, C. K., Yang, L., Cui, Y., Lin, T. Y., Belongie, S. J., & Estrin, D. (2017). Collaborative metric learning. In Proceedings of the 26th international conference on World Wide Web, Perth, Australia (pp. 193\u2013201).","DOI":"10.1145\/3038912.3052639"},{"key":"5734_CR20","doi-asserted-by":"crossref","unstructured":"Huang, K., Ying, Y., & Campbell, C. (2009). Gsml: A unified framework for sparse metric learning. In Proceedings of the 9th IEEE international conference on data mining, Miami, FL (pp. 189\u2013198).","DOI":"10.1109\/ICDM.2009.22"},{"key":"5734_CR21","unstructured":"Hwang, S. J., Grauman, K., & Sha, F. (2013). Analogy-preserving semantic embedding for visual object categorization. In Proceedings of the 30th international conference on machine learning, Atlanta, GA (pp. 639\u2013647)."},{"key":"5734_CR22","first-page":"862","volume-title":"Advances in neural information processing systems","author":"R Jin","year":"2010","unstructured":"Jin, R., Wang, S., & Zhou, Y. (2010). Regularized distance metric learning: Theory and algorithm. Advances in neural information processing systems (Vol. 23, pp. 862\u2013870). Cambridge, MA: MIT Press."},{"issue":"4","key":"5734_CR23","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1561\/2200000019","volume":"5","author":"B Kulis","year":"2012","unstructured":"Kulis, B. (2012). Metric learning: A survey. Foundations and Trends in Machine Learning, 5(4), 287\u2013364.","journal-title":"Foundations and Trends in Machine Learning"},{"issue":"1","key":"5734_CR24","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/s11263-016-0923-4","volume":"121","author":"MT Law","year":"2016","unstructured":"Law, M. T., Thome, N., & Cord, M. (2016a). Learning a distance metric from relative comparisons between quadruplets of images. International Journal of Computer Vision, 121(1), 65\u201394.","journal-title":"International Journal of Computer Vision"},{"key":"5734_CR25","doi-asserted-by":"crossref","unstructured":"Law, M. T., Yu, Y., Cord, M., & Xing, E. P. (2016b). Closed-form training of mahalanobis distance for supervised clustering. In Proceedings of the IEEE computer society conference on computer vision and pattern recognition, Las Vegas, NV (pp. 3909\u20133917)","DOI":"10.1109\/CVPR.2016.424"},{"key":"5734_CR26","unstructured":"Lim, D., Lanckriet, G., & McFee, B. (2013). Robust structural metric learning. In Proceedings of the 30th international conference on machine learning, Atlanta, GA (pp. 615\u2013623)."},{"key":"5734_CR27","first-page":"4142","volume-title":"Advances in neural information processing systems","author":"B Mason","year":"2017","unstructured":"Mason, B., Jain, L., & Nowak, R. D. (2017). Learning low-dimensional metrics. Advances in neural information processing systems (Vol. 30, pp. 4142\u20134150). Cambridge: MIT Press."},{"issue":"1","key":"5734_CR28","first-page":"148","volume":"141","author":"C McDiarmid","year":"1989","unstructured":"McDiarmid, C. (1989). On the method of bounded differences. Surveys in combinatorics, 141(1), 148\u2013188.","journal-title":"Surveys in combinatorics"},{"key":"5734_CR29","unstructured":"McFee, B., Lanckriet, G. R. (2010). Metric learning to rank. In Proceedings of the 27th international conference on machine learning, Haifa, Israel (pp.\u00a0775\u2013782)."},{"issue":"Oct","key":"5734_CR30","first-page":"839","volume":"4","author":"R Meir","year":"2003","unstructured":"Meir, R., & Zhang, T. (2003). Generalization error bounds for bayesian mixture algorithms. Journal of Machine Learning Research, 4(Oct), 839\u2013860.","journal-title":"Journal of Machine Learning Research"},{"key":"5734_CR31","first-page":"154","volume-title":"Advances in neural information processing systems","author":"M Park","year":"2015","unstructured":"Park, M., Jitkrittum, W., Qamar, A., Szab\u00f3, Z., Buesing, L., & Sahani, M. (2015). Bayesian manifold learning: The locally linear latent variable model (LL-LVM). Advances in neural information processing systems (Vol. 28, pp. 154\u2013162). Cambridge: MIT Press."},{"key":"5734_CR32","unstructured":"Perrot, M., & Habrard, A. (2015). A theoretical analysis of metric hypothesis transfer learning. In Proceedings of the 32nd international conference on machine learning, Lille, France (pp. 1708\u20131717)."},{"key":"5734_CR33","doi-asserted-by":"crossref","unstructured":"Perrot, M., Habrard, A., Muselet, D., & Sebban, M. (2014). Modeling perceptual color differences by local metric learning. In European conference on computer vision, Springer (pp. 96\u2013111).","DOI":"10.1007\/978-3-319-10602-1_7"},{"key":"5734_CR34","doi-asserted-by":"crossref","unstructured":"Qian, Q., Jin, R., Zhu, S., Lin, Y. (2015). Fine-grained visual categorization via multi-stage metric learning. In Proceedings of the IEEE computer society conference on computer vision and pattern recognition, Boston, MA (pp. 3716\u20133724).","DOI":"10.1109\/CVPR.2015.7298995"},{"issue":"3","key":"5734_CR35","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1007\/s10994-014-5456-x","volume":"99","author":"Q Qian","year":"2013","unstructured":"Qian, Q., Jin, R., Yi, J., Zhang, L., & Zhu, S. (2013). Efficient distance metric learning by adaptive sampling and mini-batch stochastic gradient descent (sgd). Machine Learning, 99(3), 353\u2013372.","journal-title":"Machine Learning"},{"issue":"May","key":"5734_CR36","first-page":"1373","volume":"13","author":"W Rejchel","year":"2012","unstructured":"Rejchel, W. (2012). On ranking and generalization bounds. Journal of Machine Learning Research, 13(May), 1373\u20131392.","journal-title":"Journal of Machine Learning Research"},{"key":"5734_CR37","doi-asserted-by":"publisher","first-page":"1104","DOI":"10.1016\/j.neucom.2015.05.013","volume":"168","author":"W Rejchel","year":"2015","unstructured":"Rejchel, W. (2015). Fast rates for ranking with large families. Neurocomputing, 168, 1104\u20131110.","journal-title":"Neurocomputing"},{"key":"5734_CR38","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107298019","volume-title":"Understanding machine learning: From theory to algorithms","author":"S Shalev-Shwartz","year":"2014","unstructured":"Shalev-Shwartz, S., & Ben-David, S. (2014). Understanding machine learning: From theory to algorithms. Cambridge: Cambridge University Press."},{"key":"5734_CR39","doi-asserted-by":"crossref","unstructured":"Shalev-Shwartz, S., Singer, Y., & Ng, A. Y. (2004). Online and batch learning of pseudo-metrics. In Proceedings of the 21st international conference on machine learning, Alberta, Canada (pp. 94\u2013102).","DOI":"10.1145\/1015330.1015376"},{"key":"5734_CR40","first-page":"2199","volume-title":"Advances in neural information processing systems","author":"N Srebro","year":"2010","unstructured":"Srebro, N., Sridharan, K., & Tewari, A. (2010). Smoothness, low noise and fast rates. Advances in neural information processing systems (pp. 2199\u20132207). Cambridge: MIT Press."},{"key":"5734_CR41","first-page":"1545","volume-title":"Advances in neural information processing systems","author":"K Sridharan","year":"2009","unstructured":"Sridharan, K., Shalev-Shwartz, S., & Srebro, N. (2009). Fast rates for regularized objectives. Advances in neural information processing systems (pp. 1545\u20131552). Cambridge: MIT Press."},{"key":"5734_CR42","first-page":"2584","volume-title":"Advances in neural information processing systems","author":"N Verma","year":"2015","unstructured":"Verma, N., & Branson, K. (2015). Sample complexity of learning mahalanobis distance metrics. Advances in neural information processing systems (Vol. 28, pp. 2584\u20132592). Cambridge: MIT Press."},{"key":"5734_CR43","first-page":"1473","volume-title":"Advances in neural information processing systems","author":"KQ Weinberger","year":"2006","unstructured":"Weinberger, K. Q., Blitzer, J., & Saul, L. K. (2006). Distance metric learning for large margin nearest neighbor classification. Advances in neural information processing systems (Vol. 18, pp. 1473\u20131480). Cambridge, MA: MIT Press."},{"key":"5734_CR44","first-page":"207","volume":"10","author":"KQ Weinberger","year":"2009","unstructured":"Weinberger, K. Q., & Saul, L. K. (2009). Distance metric learning for large margin nearest neighbor classification. Journal of Machine Learning Research, 10, 207\u2013244.","journal-title":"Journal of Machine Learning Research"},{"key":"5734_CR45","first-page":"505","volume-title":"Advances in neural information processing systems","author":"EP Xing","year":"2003","unstructured":"Xing, E. P., Ng, A. Y., Jordan, M. I., & Russell, S. (2003). Distance metric learning with application to clustering with side-information. Advances in neural information processing systems (Vol. 15, pp. 505\u2013512). Cambridge, MA: MIT Press."},{"key":"5734_CR46","unstructured":"Ye. H. J., Zhan, D. C., Si, X. M., & Jiang, Y. (2016a) Learning feature aware metric. In Proceedings of The 8th Asian conference on machine learning, Hamilton, New Zealand (pp 286\u2013301)."},{"key":"5734_CR47","first-page":"1235","volume-title":"Advances in neural information processing systems","author":"HJ Ye","year":"2016","unstructured":"Ye, H. J., Zhan, D. C., Si, X. M., Jiang, Y., & Zhou, Z. H. (2016b). What makes objects similar: A unified multi-metric learning approach. Advances in neural information processing systems (Vol. 29, pp. 1235\u20131243). Cambridge: MIT Press."},{"key":"5734_CR48","first-page":"2214","volume-title":"Advances in neural information processing systems","author":"Y Ying","year":"2009","unstructured":"Ying, Y., Huang, K., & Campbell, C. (2009). Sparse metric learning via smooth optimization. Advances in neural information processing systems (Vol. 22, pp. 2214\u20132222). Cambridge: MIT Press."},{"key":"5734_CR49","doi-asserted-by":"crossref","unstructured":"Zhan, D. C., Li, M., Li, Y. F., & Zhou, Z. H. (2009). Learning instance specific distances using metric propagation. In Proceedings of the 26th international conference on machine learning, Montreal, Canada (pp. 1225\u20131232).","DOI":"10.1145\/1553374.1553530"},{"key":"5734_CR50","unstructured":"Zhang, L., Yang, T., & Jin, R. (2017). Empirical risk minimization for stochastic convex optimization: \n                    \n                      \n                    \n                    $$O(1\/n)$$\n                    \n                      \n                        \n                          O\n                          (\n                          1\n                          \/\n                          n\n                          )\n                        \n                      \n                    \n                  -and \n                    \n                      \n                    \n                    $$O(1\/n^2)$$\n                    \n                      \n                        \n                          O\n                          (\n                          1\n                          \/\n                          \n                            n\n                            2\n                          \n                          )\n                        \n                      \n                    \n                  -type of risk bounds. In Proceedings of the 30th conference on learning theory, Amsterdam, The Netherlands (pp. 1954\u20131979)."}],"container-title":["Machine Learning"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10994-018-5734-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-018-5734-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-018-5734-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,26]],"date-time":"2019-06-26T20:03:52Z","timestamp":1561579432000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10994-018-5734-0"}},"subtitle":["Improved theoretical analysis for smooth strongly convex distance metric learning"],"short-title":[],"issued":{"date-parts":[[2018,6,27]]},"references-count":50,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,2]]}},"alternative-id":["5734"],"URL":"https:\/\/doi.org\/10.1007\/s10994-018-5734-0","relation":{},"ISSN":["0885-6125","1573-0565"],"issn-type":[{"value":"0885-6125","type":"print"},{"value":"1573-0565","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,6,27]]},"assertion":[{"value":"30 June 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 June 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 June 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}