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Entry-Wise Eigenvector Analysis and Improved Rates for Topic Modeling on Short Documents

Author

Listed:
  • Zheng Tracy Ke

    (Department of Statistics, Harvard University, Cambridge, MA 02138, USA)

  • Jingming Wang

    (Department of Statistics, Harvard University, Cambridge, MA 02138, USA)

Abstract

Topic modeling is a widely utilized tool in text analysis. We investigate the optimal rate for estimating a topic model. Specifically, we consider a scenario with n documents, a vocabulary of size p , and document lengths at the order N . When N ≥ c · p , referred to as the long-document case, the optimal rate is established in the literature at p / ( N n ) . However, when N = o ( p ) , referred to as the short-document case, the optimal rate remains unknown. In this paper, we first provide new entry-wise large-deviation bounds for the empirical singular vectors of a topic model. We then apply these bounds to improve the error rate of a spectral algorithm, Topic-SCORE. Finally, by comparing the improved error rate with the minimax lower bound, we conclude that the optimal rate is still p / ( N n ) in the short-document case.

Suggested Citation

  • Zheng Tracy Ke & Jingming Wang, 2024. "Entry-Wise Eigenvector Analysis and Improved Rates for Topic Modeling on Short Documents," Mathematics, MDPI, vol. 12(11), pages 1-41, May.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:11:p:1682-:d:1403981
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    References listed on IDEAS

    as
    1. Ruijia Wu & Linjun Zhang & T. Tony Cai, 2023. "Sparse Topic Modeling: Computational Efficiency, Near-Optimal Algorithms, and Statistical Inference," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 118(543), pages 1849-1861, July.
    2. Zheng Tracy Ke & Minzhe Wang, 2024. "Using SVD for Topic Modeling," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 119(545), pages 434-449, January.
    3. Jianqing Fan & Yingying Fan & Xiao Han & Jinchi Lv, 2022. "SIMPLE: Statistical inference on membership profiles in large networks," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(2), pages 630-653, April.
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