We present a technical survey on the state of the art approaches in data reduction and the coreset framework. These include geometric decompositions, gradient methods, random sampling, sketching and random projections. We further outline their importance for the design of streaming algorithms and give a brief overview on lower bounding techniques.

This article deals with random projections applied as a data reduction technique for Bayesian regression analysis. We show sufficient conditions under which the entire $d$-dimensional distribution is approximately preserved under random projections by reducing the number of data points from $n$ to $k\in O(\operatorname{poly}(d/\varepsilon))$ in the case $n\gg d$. Under mild assumptions, we prove that evaluating a Gaussian likelihood function based on the projected data instead of the original data yields a $(1+O(\varepsilon))$-approximation in terms of the $\ell_2$ Wasserstein distance. Our main result shows that the posterior distribution of Bayesian linear regression is approximated up to a small error depending on only an $\varepsilon$-fraction of its defining parameters. This holds when using arbitrary Gaussian priors or the degenerate case of uniform distributions over $\mathbb{R}^d$ for $\beta$. Our empirical evaluations involve different simulated settings of Bayesian linear regression. Our experiments underline that the proposed method is able to recover the regression model up to small error while considerably reducing the total running time

This paper deals with computing the smallest enclosing ball of a set of points subject to probabilistic data. In our setting, any of the n points may not or may occur at one of finitely many locations, following its own discrete probability distribution. The objective is therefore considered to be a random variable and we aim at finding a center minimizing the expected maximum distance to the points according to their distributions. Our main contribution presented in this paper is the first polynomial time (1 + ε)-approximation algorithm for the probabilistic smallest enclosing ball problem with extensions to the streaming setting.

Study Design: Retrospective analysis. Objectives: Aberrant pedicle screws can cause serious neurovascular complications. We propose that a predominant factor of pedicle screw breach is the vertebral anatomy at a given spinal level. We aim to investigate the inverse correlation between breach incidence and vertebral isthmus width. Methods: The computed tomography scans of patients undergoing thoracolumbar surgery were retrospectively reviewed. Breaches were categorized as minor (<2 mm) or major (>2 mm). Breach incidence was stratified by spinal level. Average isthmus width was then compared to the collected breach incidences. A regression analysis and Pearson’s correlation were performed. Results: A total of 656 pedicle screws were placed in 91 patients with 233 detected breaches. Incidence of major breach was 6.3%. Four patients developed post-operative radiculopathy due to breach. Breach incidence was higher in the thoracic than lumbar spine (Fisher’s exact test, P < .0001). The 2 spinal levels with the thinnest isthmus width (T4 and T5) were breached most often (73.7% and 73.9%, respectively). The 2 spinal levels with the thickest isthmus width (L4 and L5) were breached least often (20.5% and 11.8%). Breach incidence and isthmus width were shown to have a significant inverse correlation (Pearson’s correlation, R2 = 0.7, P < .0001). Conclusions: Thinner vertebral isthmus width increases pedicle screw breach incidence. Image-guided assistance may be most useful where breach incidence is highest and isthmus width is lowest (T2 to T6). Despite high incidence of cortical bone violation, there was little correlation with clinical symptoms. A breach is not automatically a clinical problem, provided the screw is structurally sound and the patient is symptomless.