Sparse Topic Modeling via Spectral Decomposition and Thresholding

In probabilistic Latent Semantic Indexing (pLSI), word frequencies across document corpora are modeled through a low-rank factorization of the expected document-term matrix into topic-word and topic-document components. In this paper, we study the estimation of the topic-word matrix under a sparsity structure motivated by Zipf's law: word frequencies within each topic exhibit a rapid empirical decay, with most probability mass concentrated on a small subset of words. Motivated by this observation, we introduce a spectral estimator that adaptively thresholds rare words prior to factorization. We show that the resulting estimator achieves an $\ell_1$-error rate whose dependence on the vocabulary size $p$ is only logarithmic. Our error bounds hold across parameter regimes, including high-dimensional settings with extremely large vocabularies, a practically important scenario that has received limited theoretical attention. Unlike many existing methods, our approach does not require the separability (or anchor-word) assumption. Synthetic and real-data experiments demonstrate that the proposed procedure is computationally efficient, statistically reliable, and effective across domains with widely varying dimensions, sparsity levels, and document lengths.