In this paper we establish an estimate for the rate of convergence of the Krasnosel'skiǐ-Mann iteration for computing fixed points of nonexpansive maps. Our main result settles the Baillon-Bruck conjecture [3] on the asymptotic regularity of this iteration. The proof proceeds by establishing a connection between these iterates and a stochastic process involving sums of non-homogeneous Bernoulli trials. We also exploit a new Hoeffding-type inequality to majorize the expected value of a convex function of these sums using Poisson distributions.

The matroid secretary problem admits several variants according to the order in which the matroid's elements are presented and how the elements are assigned weights. As the main result of this article, we devise the first constant competitive algorithm for the model in which both the order and the weight assignment are selected uniformly at random, achieving a competitive ratio of approximately 5.7187. This result is based on the nontrivial fact that every matroid can be approximately decomposed into uniformly dense minors. Based on a preliminary version of this work, Oveis Gharan and Vondrák [Proceedings of the 19th Annual European Symposium on Algorithms, ESA, 2011, pp. 335-346] devised a 40e/(e − 1)-competitive algorithm for the stronger random-assignment adversarial-order model. In this article we present an alternative algorithm achieving a competitive ratio of 16e/(e − 1). As additional results, we obtain new algorithms for the standard model of the matroid secretary problem: the adversarial-assignment random-order model. We present an O(log r)-competitive algorithm for general matroids which, unlike previous ones, uses only comparisons among seen elements. We also present constant competitive algorithms for various matroid classes, such as column-sparse representable matroids and low-density matroids. The latter class includes, as a special case, the duals of graphic matroids.

The most important open conjecture in the context of the matroid secretary problem claims the existence of an O(1)-competitive algorithm applicable to any matroid. Whereas this conjecture remains open, modified forms of it have been shown to be true, when assuming that the assignment of weights to the secretaries is not adversarial but uniformly at random [25,22]. However, so far, no variant of the matroid secretary problem with adversarial weight assignment is known that admits an O(1)-competitive algorithm. We address this point by presenting a 4-competitive procedure for the free order model, a model suggested shortly after the introduction of the matroid secretary problem, and for which no O(1)-competitive algorithm was known so far. The free order model is a relaxed version of the original matroid secretary problem, with the only difference that one can choose the order in which secretaries are interviewed.Furthermore, we consider the classical matroid secretary problem for the special case of laminar matroids. Only recently, an O(1)-competitive algorithm has been found for this case, using a clever but rather involved method and analysis [13] that leads to a competitive ratio of 16000/3. This is arguably one of the most involved special cases of the matroid secretary problem for which an O(1)-competitive algorithm is known. We present a considerably simpler and stronger 3 √ 3e ≈ 14.12-competitive procedure, based on reducing the problem to a matroid secretary problem on a partition matroid. Furthermore, our procedure is order-oblivious, which, as shown in [1], allows for transforming it into a 3 √ 3e-competitive algorithm for single-sample prophet inequalities.

Background Synthetic data may provide a solution to researchers who wish to generate and share data in support of precision healthcare. Recent advances in data synthesis enable the creation and analysis of synthetic derivatives as if they were the original data; this process has significant advantages over data deidentification. Objectives To assess a big-data platform with data-synthesizing capabilities (MDClone Ltd., Beer Sheva, Israel) for its ability to produce data that can be used for research purposes while obviating privacy and confidentiality concerns. Methods We explored three use cases and tested the robustness of synthetic data by comparing the results of analyses using synthetic derivatives to analyses using the original data using traditional statistics, machine learning approaches, and spatial representations of the data. We designed these use cases with the purpose of conducting analyses at the observation level (Use Case 1), patient cohorts (Use Case 2), and population-level data (Use Case 3). Results For each use case, the results of the analyses were sufficiently statistically similar (P > 0.05) between the synthetic derivative and the real data to draw the same conclusions. Discussion and conclusion This article presents the results of each use case and outlines key considerations for the use of synthetic data, examining their role in clinical research for faster insights and improved data sharing in support of precision healthcare.