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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. 

Motivated by modern applications in which one constructs graphical models based on a very large number of features, this paper introduces a new class of cluster-based graphical models, in which variable clustering is applied as an initial step for reducing the dimension of the feature space. We employ model assisted clustering, in which the clusters contain features that are similar to the same unobserved latent variable. Two different cluster-based Gaussian graphical models are considered: the latent variable graph, corresponding to the graphical model associated with the unobserved latent variables, and the cluster-average graph, corresponding to the vector of features averaged over clusters. Our study reveals that likelihood based inference for the latent graph, not analyzed previously, is analytically intractable. Our main contribution is the development and analysis of alternative estimation and inference strategies, for the precision matrix of an unobservable latent vector Z. We replace the likelihood of the data by an appropriate class of empirical risk functions, that can be specialized to the latent graphical model and to the simpler, but under-analyzed, cluster-average graphical model. The estimators thus derived can be used for inference on the graph structure, for instance on edge strength or pattern recovery. Inference is based on the asymptotic limits of the entry-wise estimates of the precision matrices associated with the conditional independence graphs under consideration. While taking the uncertainty induced by the clustering step into account, we establish Berry-Esseen central limit theorems for the proposed estimators. It is noteworthy that, although the clusters are estimated adaptively from the data, the central limit theorems regarding the entries of the estimated graphs are proved under the same conditions one would use if the clusters were known in advance. As an illustration of the usage of these newly developed inferential tools, we show that they can be reliably used for recovery of the sparsity pattern of the graphs we study, under FDR control, which is verified via simulation studies and an fMRI data analysis. These experimental results confirm the theoretically established difference between the two graph structures. Furthermore, the data analysis suggests that the latent variable graph, corresponding to the unobserved cluster centers, can help provide more insight into the understanding of the brain connectivity networks relative to the simpler, average-based, graph. 

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.