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On the equivalence between Non-negative Matrix Factorization and Probabilistic Latent Semantic Indexing

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  • Ding, Chris
  • Li, Tao
  • Peng, Wei

Abstract

Non-negative Matrix Factorization (NMF) and Probabilistic Latent Semantic Indexing (PLSI) have been successfully applied to document clustering recently. In this paper, we show that PLSI and NMF (with the I-divergence objective function) optimize the same objective function, although PLSI and NMF are different algorithms as verified by experiments. This provides a theoretical basis for a new hybrid method that runs PLSI and NMF alternatively, each jumping out of the local minima of the other method successively, thus achieving a better final solution. Extensive experiments on five real-life datasets show relations between NMF and PLSI, and indicate that the hybrid method leads to significant improvements over NMF-only or PLSI-only methods. We also show that at first-order approximation, NMF is identical to the [chi]2-statistic.

Suggested Citation

  • Ding, Chris & Li, Tao & Peng, Wei, 2008. "On the equivalence between Non-negative Matrix Factorization and Probabilistic Latent Semantic Indexing," Computational Statistics & Data Analysis, Elsevier, vol. 52(8), pages 3913-3927, April.
    • Handle: RePEc:eee:csdana:v:52:y:2008:i:8:p:3913-3927
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    1. 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.
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    3. Zhang, Zhong-Yuan & Gai, Yujie & Wang, Yu-Fei & Cheng, Hui-Min & Liu, Xin, 2018. "On equivalence of likelihood maximization of stochastic block model and constrained nonnegative matrix factorization," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 503(C), pages 687-697.
    4. Manini Madireddy & Ramasubramanian Sundararajan & Goda Doreswamy & Meisam Hejazi Nia & Amod Mital, 2017. "Constructing bundled offers for airline customers," Journal of Revenue and Pricing Management, Palgrave Macmillan, vol. 16(6), pages 532-552, December.
    5. Shota Saito & Yoshito Hirata & Kazutoshi Sasahara & Hideyuki Suzuki, 2015. "Tracking Time Evolution of Collective Attention Clusters in Twitter: Time Evolving Nonnegative Matrix Factorisation," PLOS ONE, Public Library of Science, vol. 10(9), pages 1-17, September.
    6. Ma, Tinghuai & Suo, Xiafei & Zhou, Jinjuan & Tang, Meili & Guan, Donghai & Tian, Yuan & Al-Dhelaan, Abdullah & Al-Rodhaan, Mznah, 2016. "Augmenting matrix factorization technique with the combination of tags and genres," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 461(C), pages 101-116.
    7. Alexandre L. M. Levada, 2021. "PCA-KL: a parametric dimensionality reduction approach for unsupervised metric learning," 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. 15(4), pages 829-868, December.
    8. Sun, Lijun & Axhausen, Kay W., 2016. "Understanding urban mobility patterns with a probabilistic tensor factorization framework," Transportation Research Part B: Methodological, Elsevier, vol. 91(C), pages 511-524.
    9. Ma, Xiaoke & Wang, Bingbo & Yu, Liang, 2018. "Semi-supervised spectral algorithms for community detection in complex networks based on equivalence of clustering methods," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 786-802.
    10. Dongjin Choi & Jun-Gi Jang & U Kang, 2019. "S3CMTF: Fast, accurate, and scalable method for incomplete coupled matrix-tensor factorization," PLOS ONE, Public Library of Science, vol. 14(6), pages 1-20, June.
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    12. Nicolas Jouvin & Pierre Latouche & Charles Bouveyron & Guillaume Bataillon & Alain Livartowski, 2021. "Greedy clustering of count data through a mixture of multinomial PCA," Computational Statistics, Springer, vol. 36(1), pages 1-33, March.
    13. Kyriaki Kalimeri & Matteo Delfino & Ciro Cattuto & Daniela Perrotta & Vittoria Colizza & Caroline Guerrisi & Clement Turbelin & Jim Duggan & John Edmunds & Chinelo Obi & Richard Pebody & Ana O Franco , 2019. "Unsupervised extraction of epidemic syndromes from participatory influenza surveillance self-reported symptoms," PLOS Computational Biology, Public Library of Science, vol. 15(4), pages 1-21, April.
    14. Travis R Meyer & Daniel Balagué & Miguel Camacho-Collados & Hao Li & Katie Khuu & P Jeffrey Brantingham & Andrea L Bertozzi, 2019. "A year in Madrid as described through the analysis of geotagged Twitter data," Environment and Planning B, , vol. 46(9), pages 1724-1740, November.

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