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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/(Nn). 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/(Nn) in the short-document case.
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Optimal Estimation of Sparse Topic Models
Topic models have become popular tools for dimension reduction and exploratory analysis of text data which consists in observed frequencies of a vocabulary of p words in n documents, stored in a p×n matrix. The main premise is that the mean of this data matrix can be factorized into a product of two non-negative matrices: a p×K word-topic matrix A and a K×n topic-document matrix W. This paper studies the estimation of A that is possibly element-wise sparse, and the number of topics K is unknown. In this under-explored context, we derive a new minimax lower bound for the estimation of such A and propose a new computationally efficient algorithm for its recovery. We derive a finite sample upper bound for our estimator, and show that it matches the minimax lower bound in many scenarios. Our estimate adapts to the unknown sparsity of A and our analysis is valid for any finite n, p, K and document lengths. Empirical results on both synthetic data and semi-synthetic data show that our proposed estimator is a strong competitor of the existing state-of-the-art algorithms for both non-sparse A and sparse A, and has superior performance is many scenarios of interest.
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- Award ID(s):
- 2015195
- PAR ID:
- 10274094
- Editor(s):
- Dalalyan, Aynak
- Date Published:
- Journal Name:
- Journal of machine learning research
- Volume:
- 21
- Issue:
- 177
- ISSN:
- 1532-4435
- Page Range / eLocation ID:
- 1 - 45
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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