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The probabilistic Latent Semantic Indexing model assumes that the expectationof the corpus matrix is low-rank and can be written as the product of atopic-word matrix and a word-document matrix. In this paper, we study theestimation of the topic-word matrix under the additional assumption that theordered entries of its columns rapidly decay to zero. This sparsity assumptionis motivated by the empirical observation that the word frequencies in a textoften adhere to Zipf's law. We introduce a new spectral procedure forestimating the topic-word matrix that thresholds words based on their corpusfrequencies, and show that its $\ell_1$-error rate under our sparsityassumption depends on the vocabulary size $p$ only via a logarithmic term. Ourerror bound is valid for all parameter regimes and in particular for thesetting where $p$ is extremely large; this high-dimensional setting is commonlyencountered but has not been adequately addressed in prior literature.Furthermore, our procedure also accommodates datasets that violate theseparability assumption, which is necessary for most prior approaches in topicmodeling. Experiments with synthetic data confirm that our procedure iscomputationally fast and allows for consistent estimation of the topic-wordmatrix in a wide variety of parameter regimes. Our procedure also performs wellrelative to well-established methods when applied to a large corpus of researchpaper abstracts, as well as the analysis of single-cell and microbiome datawhere the same statistical model is relevant but the parameter regimes arevastly different.

Sparse topic modeling via spectral decomposition and thresholding