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Clustering is a challenging problem in machine learning in which one attempts to group N objects into K0 groups based on P features measured on each object. In this article, we examine the case where N ⪠P and K0 is not known. Clustering in such high dimensional, small sample size settings has numerous applications in biology, medicine, the social sciences, clinical trials, and other scientific and experimental fields. Whereas most existing clustering algorithms either require the number of clusters to be known a priori or are sensitive to the choice of tuning parameters, our method does not require the prior specification of K0 or any tuning parameters. This represents an important advantage for our method because training data are not available in the applications we consider (i.e., in unsupervised learning problems). Without training data, estimating K0 and other hyperparameters-and thus applying alternative clustering algorithms-can be difficult and lead to inaccurate results. Our method is based on a simple transformation of the Gram matrix and application of the strong law of large numbers to the transformed matrix. If the correlation between features decays as the number of features grows, we show that the transformed feature vectors concentrate tightly around their respective cluster expectations in a low-dimensional space. This result simplifies the detection and visualization of the unknown cluster configuration. We illustrate the algorithm by applying it to 32 benchmarked microarray datasets, each containing thousands of genomic features measured on a relatively small number of tissue samples. Compared to 21 other commonly used clustering methods, we find that the proposed algorithm is faster and twice as accurate in determining the "best" cluster configuration.
RESUMO
INTRODUCTION: Case fatality among in-patients with HIV-associated tuberculosis (HIV-TB) in Africa is high. We investigated the factors associated with mortality in a rural South African hospital. METHODS: This was a prospective observational study of HIV-TB in-patients, with death by 8 weeks the endpoint. RESULTS: Of 99 patients (median CD4 count 72 cells/mm³), 32 (32%) died after median 8-day TB treatment. TB was diagnosed microbiologically in 75/99 and clinico-radiologically in 24, with no mortality difference between these groups [31% versus 38% (P = 0.53)]. Median venous lactate was 5.5 mmol/L (interquartile range 3.9-6.2) in those who died and 3.1 mmol/L (interquartile range 2.2-4.1) in survivors (P < 0.001). In multivariable analysis, lactate ≥4 mmol/L [adjusted odds ratio (aOR) 9.8, 95% confidence interval (CI): 3.0 to 32.2], Glasgow Coma Score <15 (aOR 6.6, 95% CI: 1.5 to 29.6), CD4 count <50 cells per cubic millimeter (aOR 5.5, 95% CI: 1.6 to 18.5), and age ≥50 (aOR 7.7, 95% CI: 1.2 to 46.9) independently predicted death. In a nested case-control study, comparing those who died versus CD4-matched survivors, median plasma lipopolysaccharide concentrations were 93 and 57 pg/mL (P = 0.026) and intestinal fatty acid-binding protein, 132 and 0 pg/mL (P = 0.002). CONCLUSIONS: Mortality was high and predicted by elevated lactate, likely reflecting a sepsis-syndrome secondary to TB or bacterial coinfection with intestinal barrier dysfunction appearing to contribute.
