Iterative Row Sampling

There has been significant interest and progress recently in algorithms that solve regression problems involving tall and thin matrices in input sparsity time. These algorithms find shorter equivalent of a n*d matrix where n >> d, which allows one to solve a poly(d) sized problem instead. In practice, the best performances are often obtained by invoking these routines in an iterative fashion. We show these iterative methods can be adapted to give theoretical guarantees comparable and better than the current state of the art. Our approaches are based on computing the importances of the rows, known as leverage scores, in an iterative manner. We show that alternating between computing a short matrix estimate and finding more accurate approximate leverage scores leads to a series of geometrically smaller instances. This gives an algorithm that runs in $O(nnz(A) + d^{\omega + \theta} \epsilon^{-2})$ time for any $\theta > 0$, where the $d^{\omega + \theta}$ term is comparable to the cost of solving a regression problem on the small approximation. Our results are built upon the close connection between randomized matrix algorithms, iterative methods, and graph sparsification.Comment: 26 pages, 2 figure

Different Physical Activity Subtypes and Risk of Metabolic Syndrome in Middle-Aged and Older Chinese People

Background: The prevalence of metabolic syndrome (MetS) is growing rapidly in China. Tai chi and dancing are common types of exercise among middle-aged and elderly Chinese. It remains unclear whether these activities are associated with a lower risk of MetS. Methodology/Principal Findings A total of 15,514 individuals (6,952 men, 8,562 women) aged 50 to 70 years from the Dongfeng-Tongji Cohort in Shiyan, China participated in a cross-sectional study. Physical activity and other lifestyle factors were assessed with semi-structured questionnaires during face-to-face interviews. MetS was defined by the current National Cholesterol Education Program/Adult treatment Panel III criteria for Asian Americans. The prevalence of MetS was 33.2% in the study population. In the multivariable-adjusted logistic regression analyses, total physical activity levels were monotonically associated with a lower odds of MetS [OR 0.75 comparing extreme quintiles, 95% confidence interval (CI) 0.66–0.86, P<0.001]. Compared with non-exercisers in a specific exercise type, jogging (OR 0.82, 95% CI 0.68–1.00, P = 0.046), tai chi (OR 0.72, 95% CI 0.60–0.88, P<0.001), and dancing (OR 0.56, 95% CI 0.47–0.67, P<0.001) were associated with significantly lower odds of MetS. Furthermore, each 1–h/week increment in tai chi and dancing was associated with a 5% (95% CI 2%–9%) and a 9% (95% CI 6%, 12%) lower risk of MetS. Conclusions/Significance: Jogging, tai chi and dancing are associated with a significantly lower risk of having MetS in middle-aged and older Chinese. Future intervention studies should consider the role of jogging, tai chi and dancing in preventing MetS