{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T17:57:30Z","timestamp":1767981450596,"version":"3.49.0"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T00:00:00Z","timestamp":1557705600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T00:00:00Z","timestamp":1557705600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2019,8]]},"DOI":"10.1007\/s10915-019-00971-w","type":"journal-article","created":{"date-parts":[[2019,5,14]],"date-time":"2019-05-14T02:35:56Z","timestamp":1557801356000},"page":"1139-1170","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Global Optimization with Orthogonality Constraints via Stochastic Diffusion on Manifold"],"prefix":"10.1007","volume":"80","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1844-8842","authenticated-orcid":false,"given":"Honglin","family":"Yuan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoyi","family":"Gu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rongjie","family":"Lai","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1762-0671","authenticated-orcid":false,"given":"Zaiwen","family":"Wen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,5,13]]},"reference":[{"key":"971_CR1","doi-asserted-by":"publisher","DOI":"10.1515\/9781400830244","volume-title":"Optimization Algorithms on Matrix Manifolds","author":"PA Absil","year":"2008","unstructured":"Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)"},{"key":"971_CR2","doi-asserted-by":"publisher","first-page":"4311","DOI":"10.1109\/TSP.2006.881199","volume":"54","author":"M Aharon","year":"2006","unstructured":"Aharon, M., Elad, M., Bruckstein, A.: K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation. IEEE Trans. Signal Process. 54, 4311\u20134322 (2006)","journal-title":"IEEE Trans. Signal Process."},{"key":"971_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF00941312","volume":"47","author":"F Aluffi-Pentini","year":"1985","unstructured":"Aluffi-Pentini, F., Parisi, V., Zirilli, F.: Global optimization and stochastic differential equations. J. Optim. Theory Appl. 47, 1\u201316 (1985)","journal-title":"J. Optim. Theory Appl."},{"key":"971_CR4","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1093\/imanum\/8.1.141","volume":"8","author":"J Barzilai","year":"1988","unstructured":"Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141\u2013148 (1988)","journal-title":"IMA J. Numer. Anal."},{"key":"971_CR5","doi-asserted-by":"crossref","unstructured":"Boufounos, P.T., Baraniuk, R.G.: 1-Bit compressive sensing. In: 42nd Annual Conference on Information Sciences and Systems, CISS 2008, pp. 16\u201321. IEEE (2008)","DOI":"10.1109\/CISS.2008.4558487"},{"key":"971_CR6","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1016\/j.acha.2013.10.001","volume":"37","author":"J-F Cai","year":"2014","unstructured":"Cai, J.-F., Ji, H., Shen, Z., Ye, G.-B.: Data-driven tight frame construction and image denoising. Appl. Comput. Harmon. Anal. 37, 89\u2013105 (2014)","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"971_CR7","doi-asserted-by":"publisher","first-page":"737","DOI":"10.1137\/0325042","volume":"25","author":"T-S Chiang","year":"1987","unstructured":"Chiang, T.-S., Hwang, C.-R., Sheu, S.J.: Diffusion for global optimization in $$\\mathbb{R}^n $$. SIAM J. Control Optim. 25, 737\u2013753 (1987)","journal-title":"SIAM J. Control Optim."},{"key":"971_CR8","doi-asserted-by":"publisher","first-page":"466","DOI":"10.1142\/9789814434805_0037","volume-title":"Chaos, CNN, Memristors and Beyond","author":"S-N Chow","year":"2013","unstructured":"Chow, S.-N., Yang, T.-S., Zhou, H.-M.: Global optimizations by intermittent diffusion. In: Adamatzky, A., Chen, G. (eds.) Chaos, CNN, Memristors and Beyond, pp. 466\u2013479. World Scientific, Singapore (2013)"},{"key":"971_CR9","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1137\/S0895479895290954","volume":"20","author":"A Edelman","year":"1998","unstructured":"Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constraint. SIAM J. Matrix Anal. Appl. 20, 303\u2013353 (1998)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"971_CR10","doi-asserted-by":"publisher","first-page":"1031","DOI":"10.1137\/0324060","volume":"24","author":"S Geman","year":"1986","unstructured":"Geman, S., Hwang, C.-R.: Diffusions for global optimization. SIAM J. Control Optim. 24, 1031\u20131043 (1986)","journal-title":"SIAM J. Control Optim."},{"key":"971_CR11","doi-asserted-by":"crossref","unstructured":"Gidas, B.: Global optimization via the Langevin equation. In: 24th IEEE Conference on Decision and Control, pp.\u00a0774\u2013778. IEEE (1985)","DOI":"10.1109\/CDC.1985.268602"},{"key":"971_CR12","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1137\/080726926","volume":"2","author":"D Goldfarb","year":"2009","unstructured":"Goldfarb, D., Wen, Z., Yin, W.: A curvilinear search method for p-harmonic flows on spheres. SIAM J. Imaging Sci. 2, 84\u2013109 (2009)","journal-title":"SIAM J. Imaging Sci."},{"key":"971_CR13","unstructured":"Gu, X., Yau, S.-T.: Global conformal surface parameterization. In: Proceedings of the 2003 Eurographics\/ACM SIGGRAPH Symposium on Geometry Processing, pp.\u00a0127\u2013137. Eurographics Association (2003)"},{"key":"971_CR14","volume-title":"Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations","author":"E Hairer","year":"2006","unstructured":"Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31. Springer, Berlin (2006)"},{"key":"971_CR15","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1137\/S1052623497328987","volume":"10","author":"C Helmberg","year":"2000","unstructured":"Helmberg, C., Rendl, F.: A spectral bundle method for semidefinite programming. SIAM J. Optim. 10, 673\u2013696 (2000)","journal-title":"SIAM J. Optim."},{"key":"971_CR16","doi-asserted-by":"publisher","first-page":"3445","DOI":"10.1109\/TIT.2005.855594","volume":"51","author":"O Henkel","year":"2005","unstructured":"Henkel, O.: Sphere-packing bounds in the Grassmann and Stiefel manifolds. Trans. Inf. Theory 51, 3445\u20133456 (2005)","journal-title":"Trans. Inf. Theory"},{"key":"971_CR17","volume-title":"Stochastic Analysis on Manifolds, Vol. 38 of Graduate Studies in Mathematics","author":"E Hsu","year":"2002","unstructured":"Hsu, E.: Stochastic Analysis on Manifolds, Vol. 38 of Graduate Studies in Mathematics, vol. 38. American Mathematical Society, Providence (2002)"},{"key":"971_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-14394-6","volume-title":"Stochastic Differential Equations, Universitext","author":"B\u00d8 Ksendal","year":"2003","unstructured":"Ksendal, B.\u00d8.: Stochastic Differential Equations, Universitext, 6th edn. Springer, Berlin (2003)","edition":"6"},{"key":"971_CR19","doi-asserted-by":"publisher","first-page":"705","DOI":"10.1007\/s10915-013-9752-6","volume":"58","author":"R Lai","year":"2014","unstructured":"Lai, R., Wen, Z., Yin, W., Gu, X., Lui, L.M.: Folding-free global conformal mapping for genus-0 surfaces by harmonic energy minimization. J. Sci. Comput. 58, 705\u2013725 (2014)","journal-title":"J. Sci. Comput."},{"key":"971_CR20","doi-asserted-by":"publisher","first-page":"5289","DOI":"10.1109\/TSP.2011.2162324","volume":"59","author":"JN Laska","year":"2011","unstructured":"Laska, J.N., Wen, Z., Yin, W., Baraniuk, R.G.: Trust, but verify: fast and accurate signal recovery from 1-bit compressive measurements. IEEE Trans. Signal Process. 59, 5289\u20135301 (2011)","journal-title":"IEEE Trans. Signal Process."},{"key":"971_CR21","doi-asserted-by":"publisher","first-page":"1310","DOI":"10.1137\/0726076","volume":"26","author":"S-Y Lin","year":"1989","unstructured":"Lin, S.-Y., Luskin, M.: Relaxation methods for liquid crystal problems. SIAM J. Numer. Anal. 26, 1310\u20131324 (1989)","journal-title":"SIAM J. Numer. Anal."},{"key":"971_CR22","doi-asserted-by":"publisher","first-page":"1286","DOI":"10.1137\/080729104","volume":"20","author":"C Ling","year":"2009","unstructured":"Ling, C., Nie, J., Qi, L., Ye, Y.: Biquadratic optimization over unit spheres and semidefinite programming relaxations. SIAM J. Optim. 20, 1286\u20131310 (2009)","journal-title":"SIAM J. Optim."},{"key":"971_CR23","first-page":"3","volume-title":"Lecture Notes in Computer Science","author":"X Liu","year":"2003","unstructured":"Liu, X., Srivastava, A.: Stochastic search for optimal linear representations of images on spaces with orthogonality constraints. In: Rangarajan, A., Figueiredo, M., Zerubia, J. (eds.) Lecture Notes in Computer Science, pp. 3\u201320. Springer, Berlin (2003)"},{"key":"971_CR24","unstructured":"Liu, X., Srivastava, A., Gallivan, K.: Optimal linear representations of images for object recognition. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition. Proceedings. IEEE Computer Society (2003)"},{"key":"971_CR25","doi-asserted-by":"publisher","first-page":"1758","DOI":"10.1137\/140957962","volume":"53","author":"X Liu","year":"2015","unstructured":"Liu, X., Wen, Z., Wang, X., Ulbrich, M., Yuan, Y.: On the analysis of the discretized Kohn\u2013Sham density functional theory. SIAM J. Numer. Anal. 53, 1758\u20131785 (2015)","journal-title":"SIAM J. Numer. Anal."},{"key":"971_CR26","doi-asserted-by":"publisher","first-page":"597","DOI":"10.1137\/060666743","volume":"30","author":"SJ Malham","year":"2008","unstructured":"Malham, S.J., Wiese, A.: Stochastic lie group integrators. SIAM J. Sci. Comput. 30, 597\u2013617 (2008)","journal-title":"SIAM J. Sci. Comput."},{"key":"971_CR27","first-page":"1","volume":"19","author":"PA Markowich","year":"2000","unstructured":"Markowich, P.A., Villani, C.: On the trend to equilibrium for the Fokker\u2013Planck equation: an interplay between physics and functional analysis. Mat. Contemp. 19, 1\u201329 (2000)","journal-title":"Mat. Contemp."},{"key":"971_CR28","doi-asserted-by":"publisher","first-page":"533","DOI":"10.4153\/CJM-1965-053-6","volume":"17","author":"TS Motzkin","year":"1965","unstructured":"Motzkin, T.S., Straus, E.G.: Maxima for graphs and a new proof of a theorem of Tur\u00e1n. Can. J. Math. 17, 533\u2013540 (1965)","journal-title":"Can. J. Math."},{"key":"971_CR29","doi-asserted-by":"publisher","first-page":"92","DOI":"10.1007\/BF02510919","volume":"38","author":"H Munthe-Kaas","year":"1998","unstructured":"Munthe-Kaas, H.: Runge\u2013Kutta methods on lie groups. BIT Numer. Math. 38, 92\u2013111 (1998)","journal-title":"BIT Numer. Math."},{"key":"971_CR30","volume-title":"Numerical Optimization, Springer Series in Operations Research and Financial Engineering","author":"J Nocedal","year":"2006","unstructured":"Nocedal, J., Wright, S.J.: Numerical Optimization, Springer Series in Operations Research and Financial Engineering, 2nd edn. Springer, New York (2006)","edition":"2"},{"key":"971_CR31","doi-asserted-by":"crossref","DOI":"10.1090\/clrm\/059","volume-title":"Differential Geometry and Its Applications, Classroom Resource Materials Series","author":"J Oprea","year":"2007","unstructured":"Oprea, J.: Differential Geometry and Its Applications, Classroom Resource Materials Series, 2nd edn. Mathematical Association of America, Washington (2007)","edition":"2"},{"key":"971_CR32","doi-asserted-by":"publisher","first-page":"18368","DOI":"10.1073\/pnas.1318679110","volume":"110","author":"V Ozoli\u0146\u0161","year":"2013","unstructured":"Ozoli\u0146\u0161, V., Lai, R., Caflisch, R., Osher, S.: Compressed modes for variational problems in mathematics and physics. Proc. Natl. Acad. Sci. 110, 18368\u201318373 (2013)","journal-title":"Proc. Natl. Acad. Sci."},{"key":"971_CR33","doi-asserted-by":"publisher","first-page":"461","DOI":"10.1007\/s10957-008-9473-4","volume":"141","author":"P Parpas","year":"2008","unstructured":"Parpas, P., Rustem, B.: An algorithm for the global optimization of a class of continuous minimax problems. J. Optim. Theory Appl. 141, 461\u2013473 (2008)","journal-title":"J. Optim. Theory Appl."},{"key":"971_CR34","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1007\/s10898-008-9397-4","volume":"45","author":"P Parpas","year":"2009","unstructured":"Parpas, P., Rustem, B.: Convergence analysis of a global optimization algorithm using stochastic differential equations. J. Glob. Optim. 45, 95\u2013110 (2009)","journal-title":"J. Glob. Optim."},{"key":"971_CR35","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1007\/s10898-006-9026-z","volume":"36","author":"P Parpas","year":"2006","unstructured":"Parpas, P., Rustem, B., Pistikopoulos, E.N.: Linearly constrained global optimization and stochastic differential equations. J. Glob. Optim. 36, 191\u2013217 (2006)","journal-title":"J. Glob. Optim."},{"key":"971_CR36","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1007\/s10898-007-9244-z","volume":"43","author":"P Parpas","year":"2007","unstructured":"Parpas, P., Rustem, B., Pistikopoulos, E.N.: Global optimization of robust chance constrained problems. J. Glob. Optim. 43, 231\u2013247 (2007)","journal-title":"J. Glob. Optim."},{"key":"971_CR37","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1007\/s11222-009-9120-4","volume":"20","author":"E Rubinshtein","year":"2009","unstructured":"Rubinshtein, E., Srivastava, A.: Optimal linear projections for enhancing desired data statistics. Stat. Comput. 20, 267\u2013282 (2009)","journal-title":"Stat. Comput."},{"key":"971_CR38","doi-asserted-by":"publisher","first-page":"543","DOI":"10.1137\/090767777","volume":"4","author":"A Singer","year":"2011","unstructured":"Singer, A., Shkolnisky, Y.: Three-dimensional structure determination from common lines in Cryo-EM by eigenvectors and semidefinite programming. SIAM J. Imaging Sci. 4, 543\u2013572 (2011)","journal-title":"SIAM J. Imaging Sci."},{"key":"971_CR39","unstructured":"Sloane, NJA.: Challenge problems: independent sets in graphs. https:\/\/oeis.org\/A265032\/a265032.html (2015). Accessed May 2019"},{"key":"971_CR40","doi-asserted-by":"publisher","first-page":"362","DOI":"10.1137\/0304028","volume":"4","author":"RL Stratonovich","year":"1966","unstructured":"Stratonovich, R.L.: A new representation for stochastic integrals and equations. SIAM J. Control 4, 362\u2013371 (1966)","journal-title":"SIAM J. Control"},{"key":"971_CR41","doi-asserted-by":"publisher","first-page":"701","DOI":"10.1109\/83.918563","volume":"10","author":"B Tang","year":"2001","unstructured":"Tang, B., Sapiro, G., Caselles, V.: Color image enhancement via chromaticity diffusion. IEEE Trans. Image Process. 10, 701\u2013707 (2001)","journal-title":"IEEE Trans. Image Process."},{"key":"971_CR42","doi-asserted-by":"publisher","first-page":"2085","DOI":"10.1137\/S0036142901396715","volume":"40","author":"LA Vese","year":"2002","unstructured":"Vese, L.A., Osher, S.J.: Numerical methods for p-harmonic flows and applications to image processing. SIAM J. Numer. Anal. 40, 2085\u20132104 (2002)","journal-title":"SIAM J. Numer. Anal."},{"key":"971_CR43","volume-title":"Optimal Transport, Vol. 338 of Grundlehren der mathematischen Wissenschaften","author":"C Villani","year":"2009","unstructured":"Villani, C.: Optimal Transport, Vol. 338 of Grundlehren der mathematischen Wissenschaften, vol. 338. Springer, Berlin (2009)"},{"key":"971_CR44","doi-asserted-by":"publisher","first-page":"A1299","DOI":"10.1137\/120894385","volume":"35","author":"Z Wen","year":"2013","unstructured":"Wen, Z., Milzarek, A., Ulbrich, M., Zhang, H.: Adaptive regularized self-consistent field iteration with exact Hessian for electronic structure calculation. SIAM J. Sci. Comput. 35, A1299\u2013A1324 (2013)","journal-title":"SIAM J. Sci. Comput."},{"key":"971_CR45","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1007\/s10107-012-0584-1","volume":"142","author":"Z Wen","year":"2012","unstructured":"Wen, Z., Yin, W.: A feasible method for optimization with orthogonality constraints. Math. Program. 142, 397\u2013434 (2012)","journal-title":"Math. Program."},{"key":"971_CR46","doi-asserted-by":"publisher","first-page":"529","DOI":"10.1007\/s10255-006-0328-1","volume":"22","author":"G Yin","year":"2006","unstructured":"Yin, G., Yin, K.: Global optimization using diffusion perturbations with large noise intensity. Acta Math. Appl. Sin. Engl. Ser. 22, 529\u2013542 (2006)","journal-title":"Acta Math. Appl. Sin. Engl. Ser."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-019-00971-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-019-00971-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-019-00971-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,12,12]],"date-time":"2020-12-12T17:42:59Z","timestamp":1607794979000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-019-00971-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,13]]},"references-count":46,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,8]]}},"alternative-id":["971"],"URL":"https:\/\/doi.org\/10.1007\/s10915-019-00971-w","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,13]]},"assertion":[{"value":"6 September 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 January 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 May 2019","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 May 2019","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}