1-Bit Matrix Completion

In this paper we develop a theory of matrix completion for the extreme case of noisy 1-bit observations. Instead of observing a subset of the real-valued entries of a matrix M, we obtain a small number of binary (1-bit) measurements generated according to a probability distribution determined by the real-valued entries of M. The central question we ask is whether or not it is possible to obtain an accurate estimate of M from this data. In general this would seem impossible, but we show that the maximum likelihood estimate under a suitable constraint returns an accurate estimate of M when ||M||_{\infty} <= \alpha, and rank(M) <= r. If the log-likelihood is a concave function (e.g., the logistic or probit observation models), then we can obtain this maximum likelihood estimate by optimizing a convex program. In addition, we also show that if instead of recovering M we simply wish to obtain an estimate of the distribution generating the 1-bit measurements, then we can eliminate the requirement that ||M||_{\infty} <= \alpha. For both cases, we provide lower bounds showing that these estimates are near-optimal. We conclude with a suite of experiments that both verify the implications of our theorems as well as illustrate some of the practical applications of 1-bit matrix completion. In particular, we compare our program to standard matrix completion methods on movie rating data in which users submit ratings from 1 to 5. In order to use our program, we quantize this data to a single bit, but we allow the standard matrix completion program to have access to the original ratings (from 1 to 5). Surprisingly, the approach based on binary data performs significantly better

Lyapunov dimension of elastic turbulence

Low-Reynolds-number polymer solutions exhibit a chaotic behaviour known as ‘elastic turbulence’ when the Weissenberg number exceeds a critical value. The two-dimensional Oldroyd-B model is the simplest constitutive model that reproduces this phenomenon. To make a practical estimate of the resolution scale of the dynamics, one requires the assumption that an attractor of the Oldroyd-B model exists; numerical simulations show that the quantities on which this assumption is based are bounded. We estimate the Lyapunov dimension of this assumed attractor as a function of the Weissenberg number by combining a mathematical analysis of the model with direct numerical simulations

Estimation in high dimensions: a geometric perspective

This tutorial provides an exposition of a flexible geometric framework for high dimensional estimation problems with constraints. The tutorial develops geometric intuition about high dimensional sets, justifies it with some results of asymptotic convex geometry, and demonstrates connections between geometric results and estimation problems. The theory is illustrated with applications to sparse recovery, matrix completion, quantization, linear and logistic regression and generalized linear models.Comment: 56 pages, 9 figures. Multiple minor change

Interacting for the environment:Engaging Goffman in pro-environmental action

Whilst there are profound disagreements about how more sustainable forms of living might be achieved, most research on pro-environmental action recognises it as a fundamentally social challenge – demanding shifts not merely in individuals’ attitudes and behaviours, but also in social norms, contexts and practices. Despite the social nature of the challenge, perhaps the most fundamental social medium – social interaction - remains under-theorised in this area. To address this gap, this paper applies Erving Goffman’s understandings of social interaction to an ethnographic case study of a pro- environmental change initiative called Environment Champions. The analysis shows that social interaction plays a crucially important role in shaping responses to pro-environmental change processes that has the potential to both help and hinder the spread of pro-environmental action. The paper concludes by exploring how Goffman’s ideas develop and extend current debates about pro- environmental behaviour change

M31 Pixel Lensing PLAN Campaign: MACHO Lensing and Self Lensing Signals

We present the final analysis of the observational campaign carried out by the PLAN (Pixel Lensing Andromeda) collaboration to detect a dark matter signal in form of MACHOs through the microlensing effect. The campaign consists of about 1 month/year observations carried out during 4 years (2007-2010) at the 1.5m Cassini telescope in Loiano ("Astronomical Observatory of BOLOGNA", OAB) plus 10 days of data taken in 2010 at the 2m Himalayan Chandra Telescope (HCT) monitoring the central part of M31 (two fields of about 13'x12.6'). We establish a fully automated pipeline for the search and the characterization of microlensing flux variations: as a result we detect 3 microlensing candidates. We evaluate the expected signal through a full Monte Carlo simulation of the experiment completed by an analysis of the detection efficiency of our pipeline. We consider both "self lensing" and "MACHO lensing" lens populations, given by M31 stars and dark matter halo MACHOs, in the M31 and the Milky Way (MW), respectively. The total number of events is compatible with the expected self-lensing rate. Specifically, we evaluate an expected signal of about 2 self-lensing events. As for MACHO lensing, for full 0.5 (0.01) solar mass MACHO halos, our prediction is for about 4 (7) events. The comparatively small number of expected MACHO versus self lensing events, together with the small number statistics at disposal, do not enable us to put strong constraints on that population. Rather, the hypothesis, suggested by a previous analysis, on the MACHO nature of OAB-07-N2, one of the microlensing candidates, translates into a sizeable lower limit for the halo mass fraction in form of the would be MACHO population, f, of about 15% for 0.5 solar mass MACHOs.Comment: ApJ accepted, 13 pages, 5 figures, 2 table

Care of elderly patients: a prospective audit of the prevalence of hypotension and the use of BIS intraoperatively in 25 hospitals in London

BACKGROUND: Anaesthesia is frequently complicated by intraoperative hypotension (IOH) in the elderly, and this is associated with adverse outcome. The definition of IOH is controversial, and although management guidelines for IOH in the elderly exist, the frequency of IOH and typical clinically applied treatment thresholds are largely unknown in the UK. METHODS: We audited frequency of intraoperative blood pressure against national guidelines in elderly patients undergoing surgery. Depth of anaesthesia (DOA) monitoring was also audited due to the association between low DOA values and IOH with increased mortality (as part of "double" and "triple low" phenomena) and because it is a suggested management strategy to reduce IOH. RESULTS: Twenty-five hospitals submitted data on 481 patients. Hypotension varied depending on the definition, but affected 400 patients (83.3 %) using the AAGBI standard. Furthermore, 2.9, 13.5, and 24.6 % had mean arterial blood pressures <50, <60, and <70 mmHg for 20 min, respectively, and 136 (28.4 %) had systolic blood pressure decrease by 20 % for 20 min. DOA monitors were used for 45 (9.4 %) patients. CONCLUSIONS: IOH is common and use of DOA monitors is less than implied by guidelines. Improved management of IOH may be a simple intervention with real potential to reduce morbidity in this vulnerable group

Quantization and Compressive Sensing

Quantization is an essential step in digitizing signals, and, therefore, an indispensable component of any modern acquisition system. This book chapter explores the interaction of quantization and compressive sensing and examines practical quantization strategies for compressive acquisition systems. Specifically, we first provide a brief overview of quantization and examine fundamental performance bounds applicable to any quantization approach. Next, we consider several forms of scalar quantizers, namely uniform, non-uniform, and 1-bit. We provide performance bounds and fundamental analysis, as well as practical quantizer designs and reconstruction algorithms that account for quantization. Furthermore, we provide an overview of Sigma-Delta ($\Sigma\Delta$) quantization in the compressed sensing context, and also discuss implementation issues, recovery algorithms and performance bounds. As we demonstrate, proper accounting for quantization and careful quantizer design has significant impact in the performance of a compressive acquisition system.Comment: 35 pages, 20 figures, to appear in Springer book "Compressed Sensing and Its Applications", 201