IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v58y2014i2p285-319.html
   My bibliography  Save this article

Algorithms for nonnegative matrix and tensor factorizations: a unified view based on block coordinate descent framework

Author

Listed:
  • Jingu Kim
  • Yunlong He
  • Haesun Park

Abstract

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. Copyright The Author(s) 2014

Suggested Citation

  • Jingu Kim & Yunlong He & Haesun Park, 2014. "Algorithms for nonnegative matrix and tensor factorizations: a unified view based on block coordinate descent framework," Journal of Global Optimization, Springer, vol. 58(2), pages 285-319, February.
    • Handle: RePEc:spr:jglopt:v:58:y:2014:i:2:p:285-319
    DOI: 10.1007/s10898-013-0035-4
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10898-013-0035-4
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10898-013-0035-4?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. GILLIS, Nicolas & GLINEUR, François, 2008. "Nonnegative factorization and the maximum edge biclique problem," LIDAM Discussion Papers CORE 2008064, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. GILLIS, Nicolas & GLINEUR, François, 2009. "Using underapproximations for sparse nonnegative matrix factorization," LIDAM Discussion Papers CORE 2009006, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. GILLIS, Nicolas & GLINEUR, François, 2011. "Accelerated multiplicative updates and hierarchical als algorithms for nonnegative matrix factorization," LIDAM Discussion Papers CORE 2011030, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Karthik Devarajan, 2008. "Nonnegative Matrix Factorization: An Analytical and Interpretive Tool in Computational Biology," PLOS Computational Biology, Public Library of Science, vol. 4(7), pages 1-12, July.
    5. Daniel D. Lee & H. Sebastian Seung, 1999. "Learning the parts of objects by non-negative matrix factorization," Nature, Nature, vol. 401(6755), pages 788-791, October.
    6. Berry, Michael W. & Browne, Murray & Langville, Amy N. & Pauca, V. Paul & Plemmons, Robert J., 2007. "Algorithms and applications for approximate nonnegative matrix factorization," Computational Statistics & Data Analysis, Elsevier, vol. 52(1), pages 155-173, September.
    7. M. Merritt & Y. Zhang, 2005. "Interior-Point Gradient Method for Large-Scale Totally Nonnegative Least Squares Problems," Journal of Optimization Theory and Applications, Springer, vol. 126(1), pages 191-202, July.
    8. J. Carroll & Jih-Jie Chang, 1970. "Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition," Psychometrika, Springer;The Psychometric Society, vol. 35(3), pages 283-319, September.
    9. repec:cor:louvrp:-2187 is not listed on IDEAS
    10. GILLIS, Nicolas & GLINEUR, François, 2010. "A multilevel approach for nonnegative matrix factorization," LIDAM Discussion Papers CORE 2010047, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Da Kuang & Sangwoon Yun & Haesun Park, 2015. "SymNMF: nonnegative low-rank approximation of a similarity matrix for graph clustering," Journal of Global Optimization, Springer, vol. 62(3), pages 545-574, July.
    2. Gillis, Nicolas & Glineur, François & Tuyttens, Daniel & Vandaele, Arnaud, 2015. "Heuristics for exact nonnegative matrix factorization," LIDAM Discussion Papers CORE 2015006, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Rundong Du & Barry Drake & Haesun Park, 2019. "Hybrid clustering based on content and connection structure using joint nonnegative matrix factorization," Journal of Global Optimization, Springer, vol. 74(4), pages 861-877, August.
    4. Norikazu Takahashi & Jiro Katayama & Masato Seki & Jun’ichi Takeuchi, 2018. "A unified global convergence analysis of multiplicative update rules for nonnegative matrix factorization," Computational Optimization and Applications, Springer, vol. 71(1), pages 221-250, September.
    5. Rundong Du & Da Kuang & Barry Drake & Haesun Park, 2017. "DC-NMF: nonnegative matrix factorization based on divide-and-conquer for fast clustering and topic modeling," Journal of Global Optimization, Springer, vol. 68(4), pages 777-798, August.
    6. Flavia Esposito, 2021. "A Review on Initialization Methods for Nonnegative Matrix Factorization: Towards Omics Data Experiments," Mathematics, MDPI, vol. 9(9), pages 1-17, April.
    7. Johannes Friedrich & Weijian Yang & Daniel Soudry & Yu Mu & Misha B Ahrens & Rafael Yuste & Darcy S Peterka & Liam Paninski, 2017. "Multi-scale approaches for high-speed imaging and analysis of large neural populations," PLOS Computational Biology, Public Library of Science, vol. 13(8), pages 1-24, August.
    8. Takehiro Sano & Tsuyoshi Migita & Norikazu Takahashi, 2022. "A novel update rule of HALS algorithm for nonnegative matrix factorization and Zangwill’s global convergence," Journal of Global Optimization, Springer, vol. 84(3), pages 755-781, November.
    9. Srinivas Eswar & Ramakrishnan Kannan & Richard Vuduc & Haesun Park, 2021. "ORCA: Outlier detection and Robust Clustering for Attributed graphs," Journal of Global Optimization, Springer, vol. 81(4), pages 967-989, December.
    10. Yang Qi, 2018. "A Very Brief Introduction to Nonnegative Tensors from the Geometric Viewpoint," Mathematics, MDPI, vol. 6(11), pages 1-19, October.
    11. April R. Kriebel & Joshua D. Welch, 2022. "UINMF performs mosaic integration of single-cell multi-omic datasets using nonnegative matrix factorization," Nature Communications, Nature, vol. 13(1), pages 1-17, December.
    12. Andrej Čopar & Blaž Zupan & Marinka Zitnik, 2019. "Fast optimization of non-negative matrix tri-factorization," PLOS ONE, Public Library of Science, vol. 14(6), pages 1-15, June.
    13. Duy Khuong Nguyen & Tu Bao Ho, 2017. "Accelerated parallel and distributed algorithm using limited internal memory for nonnegative matrix factorization," Journal of Global Optimization, Springer, vol. 68(2), pages 307-328, June.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Takehiro Sano & Tsuyoshi Migita & Norikazu Takahashi, 2022. "A novel update rule of HALS algorithm for nonnegative matrix factorization and Zangwill’s global convergence," Journal of Global Optimization, Springer, vol. 84(3), pages 755-781, November.
    2. Norikazu Takahashi & Ryota Hibi, 2014. "Global convergence of modified multiplicative updates for nonnegative matrix factorization," Computational Optimization and Applications, Springer, vol. 57(2), pages 417-440, March.
    3. Andrej Čopar & Blaž Zupan & Marinka Zitnik, 2019. "Fast optimization of non-negative matrix tri-factorization," PLOS ONE, Public Library of Science, vol. 14(6), pages 1-15, June.
    4. Duy Khuong Nguyen & Tu Bao Ho, 2017. "Accelerated parallel and distributed algorithm using limited internal memory for nonnegative matrix factorization," Journal of Global Optimization, Springer, vol. 68(2), pages 307-328, June.
    5. GILLIS, Nicolas & GLINEUR, François, 2011. "Accelerated multiplicative updates and hierarchical als algorithms for nonnegative matrix factorization," LIDAM Discussion Papers CORE 2011030, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. GILLIS, Nicolas & GLINEUR, François, 2008. "Nonnegative factorization and the maximum edge biclique problem," LIDAM Discussion Papers CORE 2008064, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. M. Moghadam & K. Aminian & M. Asghari & M. Parnianpour, 2013. "How well do the muscular synergies extracted via non-negative matrix factorisation explain the variation of torque at shoulder joint?," Computer Methods in Biomechanics and Biomedical Engineering, Taylor & Francis Journals, vol. 16(3), pages 291-301.
    8. Paul Fogel & Yann Gaston-Mathé & Douglas Hawkins & Fajwel Fogel & George Luta & S. Stanley Young, 2016. "Applications of a Novel Clustering Approach Using Non-Negative Matrix Factorization to Environmental Research in Public Health," IJERPH, MDPI, vol. 13(5), pages 1-14, May.
    9. Gillis, Nicolas & Glineur, François & Tuyttens, Daniel & Vandaele, Arnaud, 2015. "Heuristics for exact nonnegative matrix factorization," LIDAM Discussion Papers CORE 2015006, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    10. Jianfei Cao & Han Yang & Jianshu Lv & Quanyuan Wu & Baolei Zhang, 2023. "Estimating Soil Salinity with Different Levels of Vegetation Cover by Using Hyperspectral and Non-Negative Matrix Factorization Algorithm," IJERPH, MDPI, vol. 20(4), pages 1-15, February.
    11. FUJIWARA Yoshi & INOUE Hiroyasu & YAMAGUCHI Takayuki & AOYAMA Hideaki & TANAKA Takuma & KIKUCHI Kentaro, 2021. "Money Flow Network Among Firms' Accounts in a Regional Bank of Japan," Discussion papers 21005, Research Institute of Economy, Trade and Industry (RIETI).
    12. Flavia Esposito, 2021. "A Review on Initialization Methods for Nonnegative Matrix Factorization: Towards Omics Data Experiments," Mathematics, MDPI, vol. 9(9), pages 1-17, April.
    13. GILLIS, Nicolas & GLINEUR, François, 2010. "On the geometric interpretation of the nonnegative rank," LIDAM Discussion Papers CORE 2010051, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    14. Bastian Schaefermeier & Gerd Stumme & Tom Hanika, 2021. "Topic space trajectories," Scientometrics, Springer;Akadémiai Kiadó, vol. 126(7), pages 5759-5795, July.
    15. Yoshi Fujiwara & Rubaiyat Islam, 2021. "Bitcoin's Crypto Flow Network," Papers 2106.11446, arXiv.org, revised Jul 2021.
    16. Yin Liu & Sam Davanloo Tajbakhsh, 2023. "Stochastic Composition Optimization of Functions Without Lipschitz Continuous Gradient," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 239-289, July.
    17. Hiroyasu Abe & Hiroshi Yadohisa, 2019. "Orthogonal nonnegative matrix tri-factorization based on Tweedie distributions," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 13(4), pages 825-853, December.
    18. Rundong Du & Da Kuang & Barry Drake & Haesun Park, 2017. "DC-NMF: nonnegative matrix factorization based on divide-and-conquer for fast clustering and topic modeling," Journal of Global Optimization, Springer, vol. 68(4), pages 777-798, August.
    19. Guowei Yang & Lin Zhang & Minghua Wan, 2022. "Exponential Graph Regularized Non-Negative Low-Rank Factorization for Robust Latent Representation," Mathematics, MDPI, vol. 10(22), pages 1-20, November.
    20. José M. Maisog & Andrew T. DeMarco & Karthik Devarajan & Stanley Young & Paul Fogel & George Luta, 2021. "Assessing Methods for Evaluating the Number of Components in Non-Negative Matrix Factorization," Mathematics, MDPI, vol. 9(22), pages 1-13, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jglopt:v:58:y:2014:i:2:p:285-319. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.