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If we pick n random points uniformly in [0, 1] d and connect each point to its c d log n-nearest neighbors, where d ≥ 2 is the dimension and c d is a constant depending on the dimension, then it is well known that the graph is connected with high probability. We prove that it suffices to connect every point to c d,1 log log n points chosen randomly among its c d,2 log n-nearest neighbors to ensure a giant component of size n - o(n) with high probability. This construction yields a much sparser random graph with ~ n log log n instead of ~ n log n edges that has comparable connectivity properties. This result has nontrivial implications for problems in data science where an affinity matrix is constructed: instead of connecting each point to its k nearest neighbors, one can often pick k' ≪ k random points out of the k nearest neighbors and only connect to those without sacrificing quality of results. This approach can simplify and accelerate computation; we illustrate this with experimental results in spectral clustering of large-scale datasets.

Background: Hypertension is common in China and its prevalence is rising, yet it remains inadequately controlled. Few studies have the capacity to characterise the epidemiology and management of hypertension across many heterogeneous subgroups. We did a study of the prevalence, awareness, treatment, and control of hypertension in China and assessed their variations across many subpopulations.Methods: We made use of data generated in the China Patient-Centered Evaluative Assessment of Cardiac Events (PEACE) Million Persons Project from Sept 15, 2014, to June 20, 2017, a population-based screening project that enrolled around 1·7 million community-dwelling adults aged 35-75 years from all 31 provinces in mainland China. In this population, we defined hypertension as systolic blood pressure of at least 140 mm Hg, or diastolic blood pressure of at least 90 mm Hg, or self-reported antihypertensive medication use in the previous 2 weeks. Hypertension awareness, treatment, and control were defined, respectively, among hypertensive adults as a self-reported diagnosis of hypertension, current use of antihypertensive medication, and blood pressure of less than 140/90 mm Hg. We assessed awareness, treatment, and control in 264 475 population subgroups-defined a priori by all possible combinations of 11 demographic and clinical factors (age [35-44, 45-54, 55-64, and 65-75 years], sex [men and women], geographical region [western, central, and eastern China], urbanity [urban vs rural], ethnic origin [Han and non-Han], occupation [farmer and non-farmer], annual household income [< ¥10 000, ¥10 000-50 000, and ≥¥50 000], education [primary school and below, middle school, high school, and college and above], previous cardiovascular events [yes or no], current smoker [yes or no], and diabetes [yes or no]), and their associations with individual and primary health-care site characteristics, using mixed models.Findings: The sample contained 1 738 886 participants with a mean age of 55·6 years (SD 9·7), 59·5% of whom were women. 44·7% (95% CI 44·6-44·8) of the sample had hypertension, of whom 44·7% (44·6-44·8) were aware of their diagnosis, 30·1% (30·0-30·2) were taking prescribed antihypertensive medications, and 7·2% (7·1-7·2) had achieved control. The age-standardised and sex-standardised rates of hypertension prevalence, awareness, treatment, and control were 37·2% (37·1-37·3), 36·0% (35·8-36·2), 22·9% (22·7-23·0), and 5·7% (5·6-5·7), respectively. The most commonly used medication class was calcium-channel blockers (55·2%, 55·0-55·4). Among individuals whose hypertension was treated but not controlled, 81·5% (81·3-81·6) were using only one medication. The proportion of participants who were aware of their hypertension and were receiving treatment varied significantly across subpopulations; lower likelihoods of awareness and treatment were associated with male sex, younger age, lower income, and an absence of previous cardiovascular events, diabetes, obesity, or alcohol use (all p<0·01). By contrast, control rate was universally low across all subgroups (<30·0%).Interpretation: Among Chinese adults aged 35-75 years, nearly half have hypertension, fewer than a third are being treated, and fewer than one in twelve are in control of their blood pressure. The low number of people in control is ubiquitous in all subgroups of the Chinese population and warrants broad-based, global strategy, such as greater efforts in prevention, as well as better screening and more effective and affordable treatment.Funding: Ministry of Finance and National Health and Family Planning Commission, China. [ABSTRACT FROM AUTHOR]

Dimensionality reduction is a crucial step in essentially every single-cell RNA-sequencing (scRNA-seq) analysis. In this chapter, we describe the typical dimensionality reduction workflow that is used for scRNA-seq datasets, specifically highlighting the roles of principal component analysis, t-distributed stochastic neighborhood embedding, and uniform manifold approximation and projection in this setting. We particularly emphasize efficient computation; the software implementations used in this chapter can scale to datasets with millions of cells.

If we pick n random points uniformly in [0, 1] d and connect each point to its c d log n -nearest neighbors, where d ≥ 2 is the dimension and c d is a constant depending on the dimension, then it is well known that the graph is connected with high probability. We prove that it suffices to connect every point to c d ,1 log log n points chosen randomly among its c d ,2 log n -nearest neighbors to ensure a giant component of size n - o ( n ) with high probability. This construction yields a much sparser random graph with ~ n log log n instead of ~ n log n edges that has comparable connectivity properties. This result has nontrivial implications for problems in data science where an affinity matrix is constructed: instead of connecting each point to its k nearest neighbors, one can often pick k ' ≪ k random points out of the k nearest neighbors and only connect to those without sacrificing quality of results. This approach can simplify and accelerate computation; we illustrate this with experimental results in spectral clustering of large-scale datasets.

Recent years have witnessed intense development of randomized methods for low-rank approximation. These methods target principal component analysis and the calculation of truncated singular value decompositions. The present article presents an essentially black-box, foolproof implementation for Mathworks' MATLAB, a popular software platform for numerical computation. As illustrated via several tests, the randomized algorithms for low-rank approximation outperform or at least match the classical deterministic techniques (such as Lanczos iterations run to convergence) in basically all respects: accuracy, computational efficiency (both speed and memory usage), ease-of-use, parallelizability, and reliability. However, the classical procedures remain the methods of choice for estimating spectral norms and are far superior for calculating the least singular values and corresponding singular vectors (or singular subspaces).

Worldwide, many hundreds of thousands of stents are implanted each year to revascularize occlusions in coronary arteries. Intravascular optical coherence tomography is an important emerging imaging technique, which has the resolution and contrast necessary to quantitatively analyze stent deployment and tissue coverage following stent implantation. Automation is needed, as current, it takes up to 16 h to manually analyze hundreds of images and thousands of stent struts from a single pullback. For automated strut detection, we used image formation physics and machine learning via a Bayesian network, and 3-D knowledge of stent structure via graph search. Graph search was done on en face projections using minimum spanning tree algorithms. Depths of all struts in a pullback were simultaneously determined using graph cut. To assess the method, we employed the largest validation data set used so far, involving more than 8000 clinical images from 103 pullbacks from 72 patients. Automated strut detection achieved a 0.91±0.04 recall, and 0.84±0.08 precision. Performance was robust in images of varying quality. This method can improve the workflow for analysis of stent clinical trial data, and can potentially be used in the clinic to facilitate real-time stent analysis and visualization, aiding stent implantation.