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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. 

LOVE, a robust, scalable latent model-based clustering method for biological discovery, can be used across a range of datasets to generate both overlapping and non-overlapping clusters. In our formulation, a cluster comprises variables associated with the same latent factor and is determined from an allocation matrix that indexes our latent model. We prove that the allocation matrix and corresponding clusters are uniquely defined. We apply LOVE to biological datasets (gene expression, serological responses measured from HIV controllers and chronic progressors, vaccine-induced humoral immune responses) resulting in meaningful biological output. For all three datasets, the clusters generated by LOVE remain stable across tuning parameters. Finally, we compared LOVE's performance to that of 13 state-of-the-art methods using previously established benchmarks and found that LOVE outperformed these methods across datasets. Our results demonstrate that LOVE can be broadly used across large-scale biological datasets to generate accurate and meaningful overlapping and non-overlapping clusters.