This study uses county-level surveillance data to systematically analyze geographic variation and clustering of persons living with diagnosed HIV (PLWH) in the southern United States in 2011. Clusters corresponding to large metropolitan areas – including Miami, Atlanta, and Baltimore – had HIV prevalence rates higher (p < .001) than the regional rate. Regression analysis within the counties included in these clusters determined that race was a significant indicator for PLWH. These results provide a general picture of the distribution of PLWH in the southern United States at the county level and provide insights for identifying local geographic areas with a high number of PLWH, as well as subpopulations that may have an increased risk of infection.

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Recovery of low-rank matrices from a small number of linear measurements is now well-known to be possible under various model assumptions on the measurements. Such results demonstrate robustness and are backed with provable theoretical guarantees. However, extensions to tensor recovery have only recently began to be studied and developed, despite an abundance of practical tensor applications. Recently, a tensor variant of the Iterative Hard Thresholding method was proposed and theoretical results were obtained that guarantee exact recovery of tensors with low Tucker rank. In this paper, we utilize the same tensor version of the Restricted Isometry Property (RIP) to extend these results for tensors with low CANDECOMP/PARAFAC (CP) rank. In doing so, we leverage recent results on efficient approximations of CP decompositions that remove the need for challenging assumptions in prior works. We complement our theoretical findings with empirical results that showcase the potential of the approach.

We propose new semi-supervised nonnegative matrix factorization (SSNMF) models for document classification and provide motivation for these models as maximum likelihood estimators. The proposed SSNMF models simultaneously provide both a topic model and a model for classification, thereby offering highly interpretable classification results. We derive training methods using multiplicative updates for each new model, and demonstrate the application of these models to single-label and multi-label document classification, although the models are flexible to other supervised learning tasks such as regression. We illustrate the promise of these models and training methods on document classification datasets (e.g., 20 Newsgroups, Reuters).

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