Algorithms for nonnegative matrix and tensor factorizations: a unified view based on block coordinate descent framework
Open accessAbstract
We review algorithms developed for nonnegative matrix factorization (NMF) and nonnegative tensor factorization (NTF) from a unified view based on the block coordinate descent (BCD) framework. NMF and NTF are low-rank approximation methods for matrices and tensors in which the low-rank factors are constrained to have only nonnegative elements. The nonnegativity constraints have been shown to enable natural interpretations and allow better solutions in numerous applications including text analysis, computer vision, and bioinformatics. However, the computation of NMF and NTF remains challenging and expensive due the constraints. Numerous algorithmic approaches have been proposed to efficiently compute NMF and NTF. The BCD framework in constrained non-linear optimization readily explains the theoretical convergence properties of several efficient NMF and NTF algorithms, which are consistent with experimental observations reported in literature. In addition, we discuss algorithms that do not fit in the BCD framework contrasting them from those based on the BCD framework. With insights acquired from the unified perspective, we also propose efficient algorithms for updating NMF when there is a small change in the reduced dimension or in the data. The effectiveness of the proposed updating algorithms are validated experimentally with synthetic and real-world data sets.
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Because each subproblem is solved exactly at each iteration, the algorithm converges very quickly; previous benchmarks and the authors' own show block coordinate descent for NMF generally converges in many fewer iterations than multiplicative updates.
“Additionally, because we solve each subproblem exactly at each iteration, the algorithm converges very quickly; previous empirical benchmarks (Kim, 2014) and our own have shown that block coordinate descent algorithms for NMF generally converge in many fewer iterations than multiplicative updates (Figure S1A-B).”
From Single-Cell Multi-omic Integration Compares and Contrasts Features of Brain Cell Identity (2019) · cites this paper as Kim, 2014
The block coordinate descent algorithm satisfies the requirements of Bertsekas's theorem because each subproblem is convex with respect to the optimized block, and thus is guaranteed to converge to a fixed point (local minimum).
“As described in Kim et al, our block coordinate descent algorithm satisfies the requirements of the theorem of Bertsekas (Bertsekas, 1999), because each of the subproblems is convex with respect to the block of variables being optimized. Thus, the algorithm is guaranteed to converge to a fixed point (local minimum).”
From Single-Cell Multi-omic Integration Compares and Contrasts Features of Brain Cell Identity (2019) · cites this paper as Kim, 2014
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Cites
Gradient-based learning applied to document recognition
Lecun, Y., Bottou, L., Bengio, Y., & Haffner, P. (1998) · Proceedings of the IEEE · doi:10.1109/5.726791 · Cited by 58,805 · see citing works
Regression shrinkage and selection via the lasso
Tibshirani, R. (1996) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1996.tb02080.x · Cited by 51,482 · see citing works
Golub, G. (2012) · doi:10.56021/9781421407944 · Cited by 34,505 · see citing works
Horn, R. A., & Johnson, C. R. (1985) · Cited by 22,203 · see citing works
Learning the parts of objects by non-negative matrix factorization
Lee, D. D., & Seung, H. S. (1999) · Nature · doi:10.1038/44565 · Cited by 13,799 · see citing works
Bertsekas, D. P. (1997) · Journal of the Operational Research Society · doi:10.1038/sj.jors.2600425 · Cited by 11,969 · see citing works
Tensor Decompositions and Applications
Kolda, T. G., & Bader, B. W. (2009) · SIAM Review · doi:10.1137/07070111x · Cited by 10,595 · see citing works
ALGORITHMS FOR NON-NEGATIVE MATRIX FACTORIZATION
Lee, D. D., & Seung, H. S. (2000) · Cited by 7,972 · see citing works
Solving Least Squares Problems
Lawson, C. L., & Hanson, R. J. (1995) · doi:10.1137/1.9781611971217 · Cited by 7,103 · see citing works
Paatero, P., & Tapper, U. (1994) · Environmetrics · doi:10.1002/env.3170050203 · Cited by 6,048 · see citing works
Nonnegative Matrices in the Mathematical Sciences
Berman, A., & Plemmons, R. J. (1979) · Elsevier eBooks · doi:10.1016/c2013-0-10361-3 · Cited by 5,519 · see citing works
Atomic Decomposition by Basis Pursuit
Chen, S. S., Donoho, D. L., & Saunders, M. A. (2001) · SIAM Review · doi:10.1137/s003614450037906x · Cited by 5,030 · see citing works
Carroll, J. D., & Chang, J.-J. (1970) · Psychometrika · doi:10.1007/bf02310791 · Cited by 4,783 · see citing works
ALGORITHMS FOR NON-NEGATIVE MATRIX FACTORIZATION
Seung, D., & Lee, L.-W. (2001) · Cited by 4,753 · see citing works
Distance Metric Learning for Large Margin Nearest Neighbor Classification
Weinberger, K. Q., & Saul, L. K. (2009) · Journal of Machine Learning Research · doi:10.5555/1577069.1577078 · Cited by 4,651 · see citing works
Nonnegative Matrices in the Mathematical Sciences
(1994) · doi:10.1137/1.9781611971262 · Cited by 3,593 · see citing works
Figueiredo, M. A. T., Nowak, R. D., & Wright, S. J. (2007) · IEEE Journal of Selected Topics in Signal Processing · doi:10.1109/jstsp.2007.910281 · Cited by 3,531 · see citing works
Numerical Methods for Least Squares Problems
Björck, Å. (1996) · Society for Industrial and Applied Mathematics eBooks · doi:10.1137/1.9781611971484 · Cited by 3,276 · see citing works
Numerical methods for least squares problems
Björck, Å. (1996) · Choice Reviews Online · doi:10.5860/choice.34-1602 · Cited by 3,067 · see citing works
Solving Least Squares Problems.
Ling, R. F., Lawson, C. L., & Hanson, R. J. (1977) · Journal of the American Statistical Association · doi:10.2307/2286501 · Cited by 2,757 · see citing works
Non-negative matrix factorization with sparseness constraintsPreprint
Hoyer, P. O. (2004) · arXiv (Cornell University) · doi:10.5555/1005332.1044709 · Cited by 2,612 · see citing works
Graph Regularized Nonnegative Matrix Factorization for Data Representation
Cai, D., He, X., Han, J., & Huang, T. S. (2010) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2010.231 · Cited by 2,454 · see citing works
Harshman, R. A. (1970) · Cited by 2,292 · see citing works
Metagenes and molecular pattern discovery using matrix factorization
Brunet, J.-P., Tamayo, P., Golub, T. R., & Mesirov, J. P. (2004) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0308531101 · Cited by 2,183 · see citing works
Document clustering based on non-negative matrix factorization
Xu, W., Liu, X., & Gong, Y. (2003) · doi:10.1145/860435.860485 · Cited by 1,889 · see citing works
Projected Gradient Methods for Nonnegative Matrix Factorization
Lin, C. (2007) · Neural Computation · doi:10.1162/neco.2007.19.10.2756 · Cited by 1,768 · see citing works
Algorithms and applications for approximate nonnegative matrix factorization
Berry, M. W., Browne, M., Langville, A. N., Pauca, V. P., & Plemmons, R. J. (2006) · Computational Statistics & Data Analysis · doi:10.1016/j.csda.2006.11.006 · Cited by 1,516 · see citing works
Nonnegative Matrix and Tensor Factorizations
Cichocki, A. S., Zdunek, R., Phan, A. H., & Амари, Ш.-И. (2009) · Wiley eBooks · doi:10.1002/9780470747278 · Cited by 1,404 · see citing works
Convex and Semi-Nonnegative Matrix Factorizations
Ding, C. H. Q., Li, T., & Jordan, M. I. (2008) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2008.277 · Cited by 1,344 · see citing works
Orthogonal nonnegative matrix t-factorizations for clustering
Ding, C. H. Q., Li, T., Peng, W., & Park, H. (2006) · ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD) · doi:10.1145/1150402.1150420 · Cited by 1,273 · see citing works
Self-citationIntroduction to Global Optimization
Horst, R., Pardalos, P. M., & Thoại, N. V. (2000) · Nonconvex optimization and its applications · doi:10.1007/978-1-4615-0015-5 · Cited by 1,232 · see citing works
Févotte, C., Bertin, N., & Durrieu, J.-L. (2008) · Neural Computation · doi:10.1162/neco.2008.04-08-771 · Cited by 1,141 · see citing works
Nonmonotone Spectral Projected Gradient Methods on Convex Sets
Birgin, E. G., Martínez, J. M., & Raydan, M. (2000) · SIAM Journal on Optimization · doi:10.1137/s1052623497330963 · Cited by 1,045 · see citing works
A fast non-negativity-constrained least squares algorithm
Bro, R., & de Jong, S. (1997) · Journal of Chemometrics · doi:10.1002/(sici)1099-128x(199709/10)11:5 · Cited by 909 · see citing works
Kim, H., & Park, H. (2007) · Bioinformatics · doi:10.1093/bioinformatics/btm134 · Cited by 865 · see citing works
Self-citationLearning spatially localized, parts-based representation
Li, S. Z., Hou, X. W., Zhang, H. J., & Cheng, Q. S. (2005) · IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings - IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings · doi:10.1109/cvpr.2001.990477 · Cited by 776 · see citing works
When Does Non-Negative Matrix Factorization Give a Correct Decomposition into Parts?
Donoho, D. L., & Stodden, V. (2004) · doi:10.7916/d88d05n7 · Cited by 710 · see citing works
On the convergence of the block nonlinear Gauss–Seidel method under convex constraints
Grippo, L., & Sciandrone, M. (2000) · Operations Research Letters · doi:10.1016/s0167-6377(99)00074-7 · Cited by 706 · see citing works
On the Complexity of Nonnegative Matrix Factorization
Vavasis, S. A. (2009) · SIAM Journal on Optimization · doi:10.1137/070709967 · Cited by 601 · see citing works
Nonnegative matrix factorization for spectral data analysis
Pauca, V. P., Piper, J. W., & Plemmons, R. J. (2005) · Linear Algebra and its Applications · doi:10.1016/j.laa.2005.06.025 · Cited by 597 · see citing works
Fast Local Algorithms for Large Scale Nonnegative Matrix and Tensor Factorizations
Cichocki, A. S., & Phan, A. H. (2009) · IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences · doi:10.1587/transfun.e92.a.708 · Cited by 576 · see citing works
Kim, H., & Park, H. (2008) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/07069239x · Cited by 574 · see citing works
Self-citationNonnegative Matrix Factorization: An Analytical and Interpretive Tool in Computational Biology
Devarajan, K. (2008) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1000029 · Cited by 454 · see citing works
Symmetric Nonnegative Matrix Factorization for Graph Clustering
Kuang, D., Ding, C. H. Q., & Park, H. (2012) · SIAM International Conference on Data Mining · doi:10.1137/1.9781611972825.10 · Cited by 452 · see citing works
Self-citationUnsupervised Multiway Data Analysis: A Literature Survey
Acar, E., & Yener, B. (2008) · IEEE Transactions on Knowledge and Data Engineering · doi:10.1109/tkde.2008.112 · Cited by 451 · see citing works
On the Convergence of Multiplicative Update Algorithms for Nonnegative Matrix Factorization
Lin, C. (2007) · IEEE Transactions on Neural Networks · doi:10.1109/tnn.2007.895831 · Cited by 423 · see citing works
Generalized Nonnegative Matrix Approximations with Bregman Divergences
Sra, S., & Dhillon, I. S. (2005) · neural information processing systems · Cited by 395 · see citing works
Fast Nonnegative Matrix Factorization: An Active-Set-Like Method and Comparisons
Kim, J., & Park, H. (2011) · SIAM Journal on Scientific Computing · doi:10.1137/110821172 · Cited by 319 · see citing works
Self-citationText Mining using Non-Negative Matrix Factorizations
Pauca, V. P., Shahnaz, F., Berry, M. W., & Plemmons, R. J. (2004) · SIAM International Conference on Data Mining · doi:10.1137/1.9781611972740.45 · Cited by 311 · see citing works
Hierarchical ALS Algorithms for Nonnegative Matrix and 3D Tensor Factorization
Cichocki, A. S., Zdunek, R., & Амари, Ш.-И. (2007) · Springer eBooks · doi:10.1007/978-3-540-74494-8_22 · Cited by 291 · see citing works
Gillis, N., & Glineur, F. (2011) · Neural Computation · doi:10.1162/neco_a_00256 · Cited by 247 · see citing works
Fast algorithm for the solution of large‐scale non‐negativity‐constrained least squares problems
Benthem, M. H. V., & Keenan, M. R. (2004) · Journal of Chemometrics · doi:10.1002/cem.889 · Cited by 244 · see citing works
Fast coordinate descent methods with variable selection for non-negative matrix factorization
Hsieh, C., & Dhillon, I. S. (2011) · ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD) · doi:10.1145/2020408.2020577 · Cited by 230 · see citing works
Toward Faster Nonnegative Matrix Factorization: A New Algorithm and Comparisons
Kim, J., & Park, H. (2008) · Proceedings · doi:10.1109/icdm.2008.149 · Cited by 226 · see citing works
Self-citationSparse Nonnegative Matrix Factorization for Clustering
Kim, J., & Park, H. (2008) · Cited by 219 · see citing works
Self-citationIntroduction to Global Optimization (Nonconvex Optimization and Its Applications)
Horst, R. (2002) · Springer eBooks · Cited by 214 · see citing works
Nonnegativity constraints in numerical analysis
Chen, D., & Plemmons, R. J. (2009) · WORLD SCIENTIFIC eBooks · doi:10.1142/9789812836267_0008 · Cited by 214 · see citing works
Bayesian Non-negative Matrix Factorization
Schmidt, M. N., Winther, O., & Hansen, L. K. (2009) · Lecture notes in computer science · doi:10.1007/978-3-642-00599-2_68 · Cited by 207 · see citing works
Welling, M., & Weber, M. (2001) · Pattern Recognition Letters · doi:10.1016/s0167-8655(01)00070-8 · Cited by 175 · see citing works
Accelerating the Lee-Seung Algorithm for Nonnegative Matrix Factorization
Gonzalez, E. F., & Zhang, Y. (2005) · Rice Research Repository (Rice University) · Cited by 163 · see citing works
Position-dependent motif characterization using non-negative matrix factorization
Hutchins, L. N., Murphy, S. M., Singh, P., & Graber, J. H. (2008) · Bioinformatics · doi:10.1093/bioinformatics/btn526 · Cited by 143 · see citing works
Fast Newton-type Methods for the Least Squares Nonnegative Matrix Approximation Problem
Kim, D., Sra, S., & Dhillon, I. S. (2007) · SIAM International Conference on Data Mining · doi:10.1137/1.9781611972771.31 · Cited by 123 · see citing works
Non-negative Matrix Factorization with Quasi-Newton Optimization
Zdunek, R., & Cichocki, A. S. (2006) · Lecture notes in computer science · doi:10.1007/11785231_91 · Cited by 113 · see citing works
Using underapproximations for sparse nonnegative matrix factorization
Gillis, N., & Glineur, F. (2009) · Pattern Recognition · doi:10.1016/j.patcog.2009.11.013 · Cited by 106 · see citing works
Non-Negative Tensor Factorization using Alpha and Beta Divergences
Cichocki, A. S., Zdunek, R., Choi, S., Plemmons, R. J., & Амари, Ш.-И. (2007) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp.2007.367106 · Cited by 98 · see citing works
Frigyesi, A., & Höglund, M. (2008) · Cancer Informatics · doi:10.4137/cin.s606 · Cited by 86 · see citing works
Sequential Coordinate-Wise Algorithm for the Non-negative Least Squares Problem
Franc, V., Hlaváč, V., & Navara, M. (2005) · Lecture notes in computer science · doi:10.1007/11556121_50 · Cited by 85 · see citing works
Detect and track latent factors with online nonnegative matrix factorization
Cao, B., Shen, D., Sun, J.-T., et al. (2007) · Cited by 85 · see citing works
Computing non-negative tensor factorizations
Friedlander, M. P., & Hatz, K. (2008) · Optimization methods & software · doi:10.1080/10556780801996244 · Cited by 83 · see citing works
Interior-Point Gradient Method for Large-Scale Totally Nonnegative Least Squares Problems
Merritt, M., & Zhang, Y. (2005) · Journal of Optimization Theory and Applications · doi:10.1007/s10957-005-2668-z · Cited by 70 · see citing works
Group Sparsity in Nonnegative Matrix Factorization
Kim, J., Monteiro, R. D. C., & Park, H. (2012) · SIAM International Conference on Data Mining · doi:10.1137/1.9781611972825.73 · Cited by 70 · see citing works
Self-citationFast Nonnegative Tensor Factorization with an Active-Set-Like Method
Kim, J., & Park, H. (2012) · Springer eBooks · doi:10.1007/978-1-4471-2437-5_16 · Cited by 68 · see citing works
Self-citationZdunek, R., & Cichocki, A. S. (2008) · Computational Intelligence and Neuroscience · doi:10.1155/2008/939567 · Cited by 59 · see citing works
Nonnegative Factorization and The Maximum Edge Biclique ProblemPreprint
Gillis, N., & Glineur, F. (2008) · arXiv (Cornell University) · doi:10.48550/arxiv.0810.4225 · Cited by 58 · see citing works
Nonnegative matrix factorization : complexity, algorithms and applications
Gillis, N. (2011) · DIAL (Catholic University of Leuven) · Cited by 55 · see citing works
Bellavia, S., Macconi, M., & Morini, B. (2006) · Numerical Linear Algebra with Applications · doi:10.1002/nla.502 · Cited by 47 · see citing works
Júdice, J. J., & Pires, M. (1994) · Computers & Operations Research · doi:10.1016/0305-0548(94)90106-6 · Cited by 45 · see citing works
Fast bregman divergence NMF using taylor expansion and coordinate descent
Li, L., Lebanon, G., & Park, H. (2012) · ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD) · doi:10.1145/2339530.2339582 · Cited by 37 · see citing works
Self-citationKim, H., Park, H., & Eldén, L. (2007) · Proceedings-- IEEE International Symposium on Bioinformatics and Bioengineering · doi:10.1109/bibe.2007.4375705 · Cited by 36 · see citing works
Self-citationVideo Content Representation by Incremental Non-Negative Matrix Factorization
Bucak, S. S., & Günsel, B. (2007) · Proceedings - International Conference on Image Processing · doi:10.1109/icip.2007.4379105 · Cited by 33 · see citing works
Alternating projected Barzilai-Borwein methods for nonnegative matrix factorization.
Han, L., Neumann, M., & Prasad, U. (2009) · Cited by 32 · see citing works
Nonnegative matrix factorization via rank-one downdate
Biggs, M., Ghodsi, A., & Vavasis, S. A. (2008) · International Conference on Machine Learning · doi:10.1145/1390156.1390165 · Cited by 31 · see citing works
Low-Dimensional Polytope Approximation and Its Applications to Nonnegative Matrix Factorization
Chu, M. T., & Lin, M. M. (2008) · SIAM Journal on Scientific Computing · doi:10.1137/070680436 · Cited by 30 · see citing works
A multilevel approach for nonnegative matrix factorization
Gillis, N., & Glineur, F. (2011) · Journal of Computational and Applied Mathematics · doi:10.1016/j.cam.2011.10.002 · Cited by 30 · see citing works
On the nonnegative rank of Euclidean distance matrices
Lin, M. M., & Chu, M. T. (2010) · Linear Algebra and its Applications · doi:10.1016/j.laa.2010.03.038 · Cited by 25 · see citing works
Reversible Jump MCMC for Non-Negative Matrix Factorization
Zhong, M., & Girolami, M. (2009) · Cited by 18 · see citing works
Supervised Raman spectra estimation based on nonnegative rank deficient least squares
Drake, B., Kim, J., Mallick, M., et al. (2010) · International Conference on Information Fusion (FUSION) · doi:10.1109/icif.2010.5711882 · Cited by 12 · see citing works
Self-citationStatistical Optimization of Non-Negative Matrix Factorization
Balan, A. K., Boyles, L., Welling, M., et al. (2011) · International Conference on Artificial Intelligence and Statistics · Cited by 11 · see citing works
Block-Iterative Algorithms for Non-negative Matrix Approximation
Sra, S. (2008) · Proceedings · doi:10.1109/icdm.2008.77 · Cited by 6 · see citing works
Nonnegative matrix and tensor factorizations, least squares problems, and applications
Park, H., & Kim, J. (2011) · Medical Entomology and Zoology · Cited by 5 · see citing works
Self-citation
Flagged first, then by citations
Research works printed but not linked: 3
- Cichocki, A., Zdunek, R., Choi, S., Plemmons, R., Amari, S.-I.: Nonnegative tensor factorization using alpha and beta divergencies. In: Proceedings of the 32nd International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Honolulu, April 2007, vol. 3, pp. III-1393-III-1396 (2007)
- Friedlander, M.P., Hatz, K.: Computing nonnegative tensor factorizations. Comput. Optim. Appl. 23(4), 631-647 (2008)
- Ho, N.D.: Nonnegative matrix factorization algorithms and applications. Ph.D. thesis, Univ. Catholique de Louvain (2008)
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- Authors
- Kim, J., He, Y., & Park, H.
- Affiliations
- Georgia Institute of Technology (US)
- Corresponding author
- Park, Haesun
- Venue
- Journal of Global Optimization
- Published
- 2013-03-02
- Volume
- 58
- Issue
- 2
- Pages
- 285-319
- ISSN
- 0925-5001 (ISSN-L)
- 1573-2916
- OpenAlex
- W2144351558
- Cited by
- 352
- References
- 90
- Citations by year
- 2026: 11 · 2025: 23 · 2024: 28 · 2023: 18 · 2022: 24 · 2021: 32 · 2020: 54 · 2019: 33 · 2018: 36 · 2017: 38 · 2016: 21 · 2015: 17 · 2014: 9 · 2013: 4 · 2012: 2
- FWCI
- 5.29
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Locations
- Journal of Global Optimization (publishedVersion, cc-by)
- RePEc: Research Papers in Economics (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Computational Mathematics › Tensor decomposition and applications
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Face and Expression Recognition
- Physical Sciences › Engineering › Computational Mechanics › Sparse and Compressive Sensing Techniques
- Keywords
- non-negative matrix factorization, nonnegative tensor factorization, block coordinate descent, low-rank approximation, nonnegativity constraints, convergence analysis, dimensionality reduction, nonlinear programming
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