Orthogonal nonnegative matrix factorization problems for clustering: A new formulation and a competitive algorithm

Orthogonal Nonnegative Matrix Factorization (ONMF) with orthogonality constraints on a matrix has been found to provide better clustering results over existing clustering problems. Because of the orthogonality constraint, this optimization problem is difficult to solve. Many of the existing constrai...

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Published in:	Annals of operations research Vol. 339; no. 3; pp. 1481 - 1497
Main Authors:	Dehghanpour, Ja’far, Mahdavi-Amiri, Nezam
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.08.2024 Springer
Subjects:	Algorithms Business and Management Clustering (Computers) Combinatorics Data mining Mathematical optimization Operations Research/Decision Theory Original Research Theory of Computation Iran Isoperimetry problem Clustering Optimization problem with orthogonality constraints Orthogonal Nonnegative Matrix Factorization
ISSN:	0254-5330, 1572-9338
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Abstract	Orthogonal Nonnegative Matrix Factorization (ONMF) with orthogonality constraints on a matrix has been found to provide better clustering results over existing clustering problems. Because of the orthogonality constraint, this optimization problem is difficult to solve. Many of the existing constraint-preserving methods deal directly with the constraints using different techniques such as matrix decomposition or computing exponential matrices. Here, we propose an alternative formulation of the ONMF problem which converts the orthogonality constraints into non-convex constraints. To handle the non-convex constraints, a penalty function is applied. The penalized problem is a smooth nonlinear programming problem with quadratic (convex) constraints that can be solved by a proper optimization method. We first make use of an optimization method with two gradient projection steps and then apply a post-processing technique to construct a partition of the clustering problem. Comparative performance analysis of our proposed approach with other available clustering methods on randomly generated test problems and hard synthetic data-sets shows the outperformance of our approach, in terms of the obtained misclassification error rate and the Rand index.
AbstractList	Orthogonal Nonnegative Matrix Factorization (ONMF) with orthogonality constraints on a matrix has been found to provide better clustering results over existing clustering problems. Because of the orthogonality constraint, this optimization problem is difficult to solve. Many of the existing constraint-preserving methods deal directly with the constraints using different techniques such as matrix decomposition or computing exponential matrices. Here, we propose an alternative formulation of the ONMF problem which converts the orthogonality constraints into non-convex constraints. To handle the non-convex constraints, a penalty function is applied. The penalized problem is a smooth nonlinear programming problem with quadratic (convex) constraints that can be solved by a proper optimization method. We first make use of an optimization method with two gradient projection steps and then apply a post-processing technique to construct a partition of the clustering problem. Comparative performance analysis of our proposed approach with other available clustering methods on randomly generated test problems and hard synthetic data-sets shows the outperformance of our approach, in terms of the obtained misclassification error rate and the Rand index.
Audience	Academic
Author	Mahdavi-Amiri, Nezam Dehghanpour, Ja’far
Author_xml	– sequence: 1 givenname: Ja’far surname: Dehghanpour fullname: Dehghanpour, Ja’far organization: Faculty of Mathematical Sciences, Sharif University of Technology – sequence: 2 givenname: Nezam orcidid: 0000-0002-2763-4279 surname: Mahdavi-Amiri fullname: Mahdavi-Amiri, Nezam email: nezamm@sharif.edu organization: Faculty of Mathematical Sciences, Sharif University of Technology
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Keywords	Isoperimetry problem Clustering Optimization problem with orthogonality constraints Orthogonal Nonnegative Matrix Factorization
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References	BolteJSabachSTeboulleMProximal alternating linearized minimization for non-convex and non-smooth problemsMath Program201414645949410.1007/s10107-013-0701-9 Dehghanpour-Sahron, J., & Mahdavi-Amiri, N. (2020) A competitive optimization approach for data clustering and orthogonal non-negative matrix factorization. 4OR, 27 pages, , https://doi.org/10.1007/s10288-020-00445-y. TosyaliAKimJChoiJNew node anomaly detection algorithm based on nonnegative matrix factorization for directed citation networksAnn Oper Res202028845747410.1007/s10479-019-03508-4 YangBFuXSidiropoulosNDLearning from hidden traits: Joint factor analysis and latent clusteringIEEE Transactions on Signal Processing201765125626910.1109/TSP.2016.2614491 http://cs.joensuu.fi/sipu/datasets/. Dolan E D, & Moré J J (2002). Benchmarking optimization software with performance profiles. Mathematical Programming,91(2), 201–213. Peng, S., Ser, W., Chen, B., & Lin, Z. (2020). Robust orthogonal nonnegative matrix tri-factorization for data representation. Knowledge-Based Systems,201, 106054. FardMMThonetTGaussierEDeep k-means: Jointly clustering with k-means and learning representationsPattern Recognition Letters202013818519210.1016/j.patrec.2020.07.028 PanJNgMKOrthogonal nonnegative matrix factorization by sparsity and nuclear norm optimizationSIAM Journal on Matrix Analysis and Applications201839285687510.1137/16M1107863 Del Buono N. (2009). A penalty function for computing orthogonal non-negative matrix factorizations. (pp. 1001–1005) Facchinei, F., & Pang, J. S. (2007). Finite-dimensional variational inequalities and complementarity problems. Springer Science and Business Media. KimuraKKudoMTanakaYA column-wise update algorithm for nonnegative matrix factorization in Bregman divergence with an orthogonal constraintMachine learning2016103228530610.1007/s10994-016-5553-0 HuangSKangZXuZLiuQRobust deep k-means: An effective and simple method for data clusteringPattern Recognition202111710799610.1016/j.patcog.2021.107996 LancichinettiAFortunatoSBenchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communitiesPhysical Review E20098001611810.1103/PhysRevE.80.016118 Ding, C., Li, T., Peng, W., & Park, H. (2006). Orthogonal nonnegative matrix t-factorizations for clustering. (pp. 126–135) ShefiRTeboulleMOn the rate of convergence of the proximal alternating linearized minimization algorithm for convex problemsEURO J Comput Optim20164274610.1007/s13675-015-0048-5 FräntiPSieranojaSK-means properties on six clustering benchmark datasetsApplied Intelligence201848124743475910.1007/s10489-018-1238-7 McQueenJSome methods for classification and analysis of multivariate observationsComputer and Chemistry19674257272 DinlerDTuralMKOzdemirelNECentroid based Tree-Structured Data Clustering Using Vertex/Edge Overlap and Graph Edit DistanceAnn Oper Res202028918512210.1007/s10479-019-03505-7 XiaSPengDMengDZhangCWangGGiemEChenZA fast adaptive k-means with no boundsIEEE Transactions on Pattern Analysis and Machine Intelligence202010.1109/TPAMI.2020.3008694 BertsekasDPNonlinear Programming19992Belmont, MassachusettsAthena Scientific YuSSChuSWWangCMChanYKChangTCTwo improved k-means algorithmsApplied Soft Computing20186874775510.1016/j.asoc.2017.08.032 PaateroPTapperUPositive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data valuesEnvironmetrics19945211112610.1002/env.3170050203 Arthur, D., Sergi, V. (2007) K-means++: The Advantages of Careful Seeding. SODA ’ 07: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 1027-1035 . BankerRDChangHZhengZOn the use of super-efficiency procedures for ranking efficient units and identifying outliersAnn Oper Res20172501213510.1007/s10479-015-1980-8 KimJParkHFast non-negative matrix factorization: An active-set-like method and comparisonsSIAM Journal on Scientific Computing20113363261328110.1137/110821172 LawrenceHPhippsAComparing partitionsJournal of Classification19852119321810.1007/BF01908075 Ng, A. Y., Jordan, M. I., & Weiss, Y (2002) On spectral clustering: analysis and an algorithm. In: Advances in Neural Information Processing Systems, 849-856 . DaneshgarAJavadiRRazaviSSClustering and outlier detection using isoperimetric number of treesPattern Recognition201346123371338210.1016/j.patcog.2013.05.015 SinagaKPYangMSUnsupervised K-means clustering algorithm. 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Pock, T., & SabachS. (2016). Inertial proximal alternating linearized minimization (iPALM) for nonconvex and nonsmooth problems. SIAM Journal on Imaging Sciences,9(4), 1756–1787. Kimura, K., Tanaka, Y., & Kudo, M. (2015). A fast hierarchical alternating least squares algorithm for orthogonal nonnegative matrix factorization. JiangBDaiYHA framework of constraint preserving update schemes for optimization on Stiefel manifoldMathematical Programming2015153253557510.1007/s10107-014-0816-7 LiWLiJLiuXDongLTwo fast vector-wise update algorithms for orthogonal nonnegative matrix factorization with sparsity constraintJournal of Computational and Applied Mathematics202037511278510.1016/j.cam.2020.112785 DuanLXuLLiuYCluster-based outlier detection. 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References_xml	– reference: Ding, C., Li, T., Peng, W., & Park, H. (2006). Orthogonal nonnegative matrix t-factorizations for clustering. (pp. 126–135) – reference: http://cs.joensuu.fi/sipu/datasets/. – reference: QinZWanTZhaoHHybrid clustering of data and vague concepts based on labels semanticsAnn Oper Res2017256239341610.1007/s10479-017-2541-0 – reference: HuangSKangZXuZLiuQRobust deep k-means: An effective and simple method for data clusteringPattern Recognition202111710799610.1016/j.patcog.2021.107996 – reference: BankerRDChangHZhengZOn the use of super-efficiency procedures for ranking efficient units and identifying outliersAnn Oper Res20172501213510.1007/s10479-015-1980-8 – reference: ShefiRTeboulleMOn the rate of convergence of the proximal alternating linearized minimization algorithm for convex problemsEURO J Comput Optim20164274610.1007/s13675-015-0048-5 – reference: BertsekasDPNonlinear Programming19992Belmont, MassachusettsAthena Scientific – reference: Dolan E D, & Moré J J (2002). Benchmarking optimization software with performance profiles. Mathematical Programming,91(2), 201–213. – reference: McQueenJSome methods for classification and analysis of multivariate observationsComputer and Chemistry19674257272 – reference: BolteJSabachSTeboulleMProximal alternating linearized minimization for non-convex and non-smooth problemsMath Program201414645949410.1007/s10107-013-0701-9 – reference: Arthur, D., Sergi, V. (2007) K-means++: The Advantages of Careful Seeding. SODA ’ 07: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 1027-1035 . – reference: TosyaliAKimJChoiJNew node anomaly detection algorithm based on nonnegative matrix factorization for directed citation networksAnn Oper Res202028845747410.1007/s10479-019-03508-4 – reference: KimJParkHFast non-negative matrix factorization: An active-set-like method and comparisonsSIAM Journal on Scientific Computing20113363261328110.1137/110821172 – reference: FräntiPSieranojaSK-means properties on six clustering benchmark datasetsApplied Intelligence201848124743475910.1007/s10489-018-1238-7 – reference: YuSSChuSWWangCMChanYKChangTCTwo improved k-means algorithmsApplied Soft Computing20186874775510.1016/j.asoc.2017.08.032 – reference: FardMMThonetTGaussierEDeep k-means: Jointly clustering with k-means and learning representationsPattern Recognition Letters202013818519210.1016/j.patrec.2020.07.028 – reference: LiWLiJLiuXDongLTwo fast vector-wise update algorithms for orthogonal nonnegative matrix factorization with sparsity constraintJournal of Computational and Applied Mathematics202037511278510.1016/j.cam.2020.112785 – reference: DuanLXuLLiuYCluster-based outlier detection. AnnOper Res200916815116810.1007/s10479-008-0371-9 – reference: PompiliFGillisNAbsilPAGlineurFTwo algorithms for orthogonal non-negative matrix factorization with application to clusteringNeurocomputing2014141152510.1016/j.neucom.2014.02.018 – reference: XiaSPengDMengDZhangCWangGGiemEChenZA fast adaptive k-means with no boundsIEEE Transactions on Pattern Analysis and Machine Intelligence202010.1109/TPAMI.2020.3008694 – reference: http://www.vision.caltech.edu/lihi/Demos/SelfTuningClustering.html. – reference: Dehghanpour-Sahron, J., & Mahdavi-Amiri, N. (2020) A competitive optimization approach for data clustering and orthogonal non-negative matrix factorization. 4OR, 27 pages, , https://doi.org/10.1007/s10288-020-00445-y. – reference: PanJNgMKOrthogonal nonnegative matrix factorization by sparsity and nuclear norm optimizationSIAM Journal on Matrix Analysis and Applications201839285687510.1137/16M1107863 – reference: PaateroPTapperUPositive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data valuesEnvironmetrics19945211112610.1002/env.3170050203 – reference: Facchinei, F., & Pang, J. S. (2007). Finite-dimensional variational inequalities and complementarity problems. Springer Science and Business Media. – reference: Kimura, K., Tanaka, Y., & Kudo, M. (2015). A fast hierarchical alternating least squares algorithm for orthogonal nonnegative matrix factorization. – reference: MorenoSPereiraJYushimitoWA hybrid K-means and integer programming method for commercial territory design: a case study in meat distributionAnn Oper Res202028618711710.1007/s10479-017-2742-6 – reference: Ng, A. Y., Jordan, M. I., & Weiss, Y (2002) On spectral clustering: analysis and an algorithm. In: Advances in Neural Information Processing Systems, 849-856 . – reference: DinlerDTuralMKOzdemirelNECentroid based Tree-Structured Data Clustering Using Vertex/Edge Overlap and Graph Edit DistanceAnn Oper Res202028918512210.1007/s10479-019-03505-7 – reference: JiangBDaiYHA framework of constraint preserving update schemes for optimization on Stiefel manifoldMathematical Programming2015153253557510.1007/s10107-014-0816-7 – reference: Del Buono N. (2009). A penalty function for computing orthogonal non-negative matrix factorizations. 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Snippet	Orthogonal Nonnegative Matrix Factorization (ONMF) with orthogonality constraints on a matrix has been found to provide better clustering results over existing...
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SubjectTerms	Algorithms Business and Management Clustering (Computers) Combinatorics Data mining Mathematical optimization Operations Research/Decision Theory Original Research Theory of Computation
Title	Orthogonal nonnegative matrix factorization problems for clustering: A new formulation and a competitive algorithm
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