Topic modeling is a useful tool in computational social science, digital humanities, biology, and chemistry. A popular topic model is the probabilistic Latent Semantic Indexing (pLSI) model. It assumes that the word-document matrix factorizes into the product of a low-rank word-topic matrix A, and a low-rank topic-document matrixW. The goal is to estimate these matrices. While many algorithms are available for topic modeling, there is relatively little statistical understanding. The first contribution of this thesis is providing rigorous statistical theory for both problems, including the optimal rate of convergence for estimating A, the optimal rate of convergence for estimating W, and an unconventional theory for including "sparsity" in topic modeling. The second contribution of this thesis is proposing an assortment of new methods, including a spectral approach to estimating A, a spectral approach to estimating W, and a word-screening method. All these methods are computationally efficient and statistically optimal for a wide range of settings. The thesis is composed of three parts. In the first part we propose a new algorithm for estimating the word-topic matrix using the entry-wise ratios of the left singular vectors of the normalized word-document matrix, which is shown to possess the minimax optimal row-wise error rate using an entry-wise bounds for singular vectors. The second part we study topic-document matrix estimation problem, where we introduce a new notion of sparsity, the non-informativeness, and propose to use a new non-informative words screening method, before conducting topic-document matrix estimation based on the resulting right singular vectors of the normalized word-document matrix. We show the algorithm enjoys minimax convergence rate under the existence of the non-informative words. Both algorithms are simple, but surprisingly enjoy various deep algebraic insights underneath the pLSI model. In the last part we study a different but topic-model-related problem, information retrieval, where we propose a new model-based algorithm which explicitly takes into account the heterogeneity between documents and queries generation. In each part we also provide various simulations and real data applications to support the competitiveness of our proposed models and algorithms.




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