Optimal query complexity for estimating the trace of a matrix

Given an implicit $n\times n$ matrix $A$ with oracle access $x^TA x$ for any $x\in \mathbb{R}^n$, we study the query complexity of randomized algorithms for estimating the trace of the matrix. This problem has many applications in quantum physics, machine learning, and pattern matching. Two metrics are commonly used for evaluating the estimators: i) variance; ii) a high probability multiplicative-approximation guarantee. Almost all the known estimators are of the form $\frac{1}{k}\sum_{i=1}^k x_i^T A x_i$ for $x_i\in \mathbb{R}^n$ being i.i.d. for some special distribution. Our main results are summarized as follows. We give an exact characterization of the minimum variance unbiased estimator in the broad class of linear nonadaptive estimators (which subsumes all the existing known estimators). We also consider the query complexity lower bounds for any (possibly nonlinear and adaptive) estimators: (1) We show that any estimator requires $\Omega(1/\epsilon)$ queries to have a guarantee of variance at most $\epsilon$. (2) We show that any estimator requires $\Omega(\frac{1}{\epsilon^2}\log \frac{1}{\delta})$ queries to achieve a $(1\pm\epsilon)$-multiplicative approximation guarantee with probability at least $1 - \delta$. Both above lower bounds are asymptotically tight. As a corollary, we also resolve a conjecture in the seminal work of Avron and Toledo (Journal of the ACM 2011) regarding the sample complexity of the Gaussian Estimator.Comment: full version of the paper in ICALP 201

Can pervasive sensing address current challenges in global healthcare?

Important challenges facing global healthcare include the increase in the number of people affected by escalating healthcare costs, chronic and infectious diseases, the need for better and more affordable elderly care and expanding urbanisation combined with air and water pollution. Recent advances in pervasive sensing technologies have led to miniaturised sensor networks that can be worn or integrated within the living environment without affecting a person's daily patterns. These sensors promise to change healthcare from snapshot measurements of physiological parameters to continuous monitoring enabling clinicians to provide guidance on a daily basis. This article surveys several of the solutions provided by these sensor platforms from elderly care to neonatal monitoring and environmental mapping. Some of the opportunities available and the challenges facing the adoption of such technologies in large-scale epidemiological studies are also discussed

A Useful Form of the Abel Bound and Its Application to Estimator Threshold Prediction

International audienceThis correspondence investigates the Abel bound in order to predict the estimators mean square error (mse) threshold effect. A tractable and computationally efficient form of this bound is derived. This form combines the Chapman–Robbins and the Cramér–Rao bounds. This bound is applied to a data-aided carrier frequency estimation problem for which a closed-form expression is provided. An indicator of the signal-to-noise ratio threshold is proposed. A comparison with recent results on the Barankin bound (Chapman–Robbins version) shows the superiority of the Abel-bound version to predict the mse threshold without increasing the computational complexity

The effect of Duffing-type non-linearities and Coulomb damping on the response of an energy harvester to random excitations

Linear energy harvesters can only produce useful amounts of power when excited close to their natural frequency. Due to the uncertain nature of ambient vibrations, it has been hypothesised that such devices will perform poorly in real-world applications. To improve performance, it has been suggested that the introduction of non-linearities into such devices may extend the bandwidth over which they perform effectively. In this study, a magnetic levitation device with non-linearities similar to the Duffing oscillator is considered. The governing equations of the device are formed in which the effects of friction are considered. Analytical solutions are used to explore the effect that friction can have on the system when it is under harmonic excitations. Following this, a numerical model is formed. A differential evolution algorithm is used alongside experimental data to identify the relevant parameters of the device. The model is then validated using experimental data. Monte Carlo simulations are then used to analyse the effect of coulomb damping and Duffing-type non-linearities when the device is subjected to broadband white noise and coloured noise excitations. </jats:p

Calcium supplementation to prevent pre-eclampsia - a systematic review

Background. Calcium supplementation during pregnancy may prevent high blood pressure and preterm labour. Objective. To assess the effects of calcium supplementation during pregnancy on hypertensive disorders of pregnancy and related maternal and child adverse outcomes. Design. A systematic review of randomised trials that compared supplementation with at least 1 g calcium daily during pregnancy with placebo. Search strategy. The Cochrane Pregnancy and Childbirth Group trials register (October 2001) and the Cochrane Controlled Trials Register (Issue 3, 2001) were searched and study s were contacted. Data collection and analysis. Eligibility and trial quality were assessed. Data were extracted and analysed. Main results. There was a modest reduction in the risk of pre-eclampsia with calcium supplementation (relative risk (RR) 0.68, 95% confidence interval (CI): 0.57- 0.81). The effect was greatest for women at high risk of hypertension (RR 0.21, 95% CI: 0.11 - 0.39) and those with low baseline calcium intake (RR 0.32, 95% CI: 0.21 - 0.49). There was no overall effect on the risk of preterm delivery, although there was a reduction in risk among women at high risk of hypertension (RR 0.42, 95% CI: 0.23 - 0.78). There was no evidence of any effect of calcium supplementation on stillbirth or death before discharge from hospital. There were fewer babies with birthweight 95th percentile was reduced (RR 0.59, 95% CI: 0.39 - 0.91). Conclusions. Calcium supplementation appears to be beneficial for women at high risk of gestational hypertension and in communities with low dietary calcium intake. These benefits were confined to several rather small trials, and were not found in the largest trial to date, conducted in a low-risk population. Further research is required. (South African Medical Journal: 2003 93(3): 224-228

Prenatal Diagnosis of Gómez-López-Hernández Syndrome.

Gómez-López-Hernández syndrome (GLHS), also known as cerebello-trigeminal-dermal dysplasia, is an extremely rare neurocutaneous disease, classically described by the triad of rhombencephalosynapsis (RES), bilateral focal alopecia, and trigeminal anesthesia. The clinical and radiographic spectrum of GLHS is now known to be broader, including craniofacial and supratentorial anomalies, as well as neurodevelopmental issues. Here, we present a case of antenatally diagnosed GLHS with RES, hydrocephaly, and craniofacial anomalies identified on ultrasound (low-set ears with posterior rotation, hypertelorism, midface hypoplasia, micrognathia, and anteverted nares) which were confirmed by autopsy after termination of pregnancy at 23 weeks of gestation. As no known genetic causes have been identified and the classical triad is not applicable to prenatal imaging, prenatal diagnosis of GLHS is based on neuroimaging and the identification of supporting features. In presence of an RES associated with craniofacial abnormalities in prenatal (brachycephaly, turricephaly, low-set ears, midface retrusion, micrognathia), GLHS should be considered as "possible" according to postnatal criteria