Support Vector Machines and Radon's Theorem

A support vector machine (SVM) is an algorithm which finds a hyperplane that optimally separates labeled data points in $\mathbb{R}^n$ into positive and negative classes. The data points on the margin of this separating hyperplane are called support vectors. We study the possible configurations of support vectors for points in general position. In particular, we connect the possible configurations to Radon's theorem, which provides guarantees for when a set of points can be divided into two classes (positive and negative) whose convex hulls intersect. If the positive and negative support vectors in a generic SVM configuration are projected to the separating hyperplane, then these projected points will form a Radon configuration. Further, with a particular type of general position, we show there are at most $n+1$ support vectors. This can be used to test the level of machine precision needed in a support vector machine implementation. We also show the projections of the convex hulls of the support vectors intersect in a single Radon point, and under a small enough perturbation, the points labeled as support vectors remain labeled as support vectors. We furthermore consider computations studying the expected number of support vectors for randomly generated data

A GPU-Oriented Algorithm Design for Secant-Based Dimensionality Reduction

Dimensionality-reduction techniques are a fundamental tool for extracting useful information from high-dimensional data sets. Because secant sets encode manifold geometry, they are a useful tool for designing meaningful data-reduction algorithms. In one such approach, the goal is to construct a projection that maximally avoids secant directions and hence ensures that distinct data points are not mapped too close together in the reduced space. This type of algorithm is based on a mathematical framework inspired by the constructive proof of Whitney's embedding theorem from differential topology. Computing all (unit) secants for a set of points is by nature computationally expensive, thus opening the door for exploitation of GPU architecture for achieving fast versions of these algorithms. We present a polynomial-time data-reduction algorithm that produces a meaningful low-dimensional representation of a data set by iteratively constructing improved projections within the framework described above. Key to our algorithm design and implementation is the use of GPUs which, among other things, minimizes the computational time required for the calculation of all secant lines. One goal of this report is to share ideas with GPU experts and to discuss a class of mathematical algorithms that may be of interest to the broader GPU community.Comment: To appear in the 17th IEEE International Symposium on Parallel and Distributed Computing, Geneva, Switzerland 201

A SHAPE-CONTEXT MODEL FOR MATCHING PLACENTAL CHORIONIC SURFACE VASCULAR NETWORKS

Placental chorionic surface vascular networks (PCSVNs) are essential high-capacitance, low-resistance distribution and drainage networks, and are hence important to placental function and to fetal and newborn health. It was hypothesized that variations in the PCSVN structure may reflect both the overall effects of genetic and environmentally regulated variations in branching morphogenesis within the conceptus and the fetus’s vital organs. A critical step in PCSVN analysis is the extraction of blood vessel structure, which has only been done manually through a laborious process, making studies in large cohorts and applications in clinical settings nearly impossible. The large variation in the shape, color, and texture of the placenta presents significant challenges to both machine and human to accurately extract PCSVNs. To increase the visibility of the vessels, colored paint can be injected into the vascular networks of placentas, allowing PCSVNs to be manually traced with a high level of accuracy. This paper provides a proof-of-concept study to explain the geometric differences between manual tracings of paint-injected and un-manipulated PCSVNs under the framework of a shape-context model. Under this framework, paint-injected and un-manipulated tracings of PCSVNs can be matched with nearly 100% accuracy. The implication of our results is that the manual tracing protocol yields faithful PCSVN representations modulo a set of affine transformations, making manual tracing a reliable method for studying PCSVNs. Our work provides assurance to a new pre-processing approach for studying vascular networks by ways of dye-injection in medical imaging problems

Prevalence, associated factors and outcomes of pressure injuries in adult intensive care unit patients: the DecubICUs study

Funder: European Society of Intensive Care Medicine; doi: http://dx.doi.org/10.13039/501100013347Funder: Flemish Society for Critical Care NursesAbstract: Purpose: Intensive care unit (ICU) patients are particularly susceptible to developing pressure injuries. Epidemiologic data is however unavailable. We aimed to provide an international picture of the extent of pressure injuries and factors associated with ICU-acquired pressure injuries in adult ICU patients. Methods: International 1-day point-prevalence study; follow-up for outcome assessment until hospital discharge (maximum 12 weeks). Factors associated with ICU-acquired pressure injury and hospital mortality were assessed by generalised linear mixed-effects regression analysis. Results: Data from 13,254 patients in 1117 ICUs (90 countries) revealed 6747 pressure injuries; 3997 (59.2%) were ICU-acquired. Overall prevalence was 26.6% (95% confidence interval [CI] 25.9–27.3). ICU-acquired prevalence was 16.2% (95% CI 15.6–16.8). Sacrum (37%) and heels (19.5%) were most affected. Factors independently associated with ICU-acquired pressure injuries were older age, male sex, being underweight, emergency surgery, higher Simplified Acute Physiology Score II, Braden score 3 days, comorbidities (chronic obstructive pulmonary disease, immunodeficiency), organ support (renal replacement, mechanical ventilation on ICU admission), and being in a low or lower-middle income-economy. Gradually increasing associations with mortality were identified for increasing severity of pressure injury: stage I (odds ratio [OR] 1.5; 95% CI 1.2–1.8), stage II (OR 1.6; 95% CI 1.4–1.9), and stage III or worse (OR 2.8; 95% CI 2.3–3.3). Conclusion: Pressure injuries are common in adult ICU patients. ICU-acquired pressure injuries are associated with mainly intrinsic factors and mortality. Optimal care standards, increased awareness, appropriate resource allocation, and further research into optimal prevention are pivotal to tackle this important patient safety threat

Correction to: Prevalence, associated factors and outcomes of pressure injuries in adult intensive care unit patients: the DecubICUs study

The original version of this article unfortunately contained a mistake

Prevalence, associated factors and outcomes of pressure injuries in adult intensive care unit patients: the DecubICUs study

Funder: European Society of Intensive Care Medicine; doi: http://dx.doi.org/10.13039/501100013347Funder: Flemish Society for Critical Care NursesAbstract: Purpose: Intensive care unit (ICU) patients are particularly susceptible to developing pressure injuries. Epidemiologic data is however unavailable. We aimed to provide an international picture of the extent of pressure injuries and factors associated with ICU-acquired pressure injuries in adult ICU patients. Methods: International 1-day point-prevalence study; follow-up for outcome assessment until hospital discharge (maximum 12 weeks). Factors associated with ICU-acquired pressure injury and hospital mortality were assessed by generalised linear mixed-effects regression analysis. Results: Data from 13,254 patients in 1117 ICUs (90 countries) revealed 6747 pressure injuries; 3997 (59.2%) were ICU-acquired. Overall prevalence was 26.6% (95% confidence interval [CI] 25.9–27.3). ICU-acquired prevalence was 16.2% (95% CI 15.6–16.8). Sacrum (37%) and heels (19.5%) were most affected. Factors independently associated with ICU-acquired pressure injuries were older age, male sex, being underweight, emergency surgery, higher Simplified Acute Physiology Score II, Braden score 3 days, comorbidities (chronic obstructive pulmonary disease, immunodeficiency), organ support (renal replacement, mechanical ventilation on ICU admission), and being in a low or lower-middle income-economy. Gradually increasing associations with mortality were identified for increasing severity of pressure injury: stage I (odds ratio [OR] 1.5; 95% CI 1.2–1.8), stage II (OR 1.6; 95% CI 1.4–1.9), and stage III or worse (OR 2.8; 95% CI 2.3–3.3). Conclusion: Pressure injuries are common in adult ICU patients. ICU-acquired pressure injuries are associated with mainly intrinsic factors and mortality. Optimal care standards, increased awareness, appropriate resource allocation, and further research into optimal prevention are pivotal to tackle this important patient safety threat