Unicity conditions for low-rank matrix recovery

Low-rank matrix recovery addresses the problem of recovering an unknown low-rank matrix from few linear measurements. Nuclear-norm minimization is a tractible approach with a recent surge of strong theoretical backing. Analagous to the theory of compressed sensing, these results have required random measurements. For example, m >= Cnr Gaussian measurements are sufficient to recover any rank-r n x n matrix with high probability. In this paper we address the theoretical question of how many measurements are needed via any method whatsoever --- tractible or not. We show that for a family of random measurement ensembles, m >= 4nr - 4r^2 measurements are sufficient to guarantee that no rank-2r matrix lies in the null space of the measurement operator with probability one. This is a necessary and sufficient condition to ensure uniform recovery of all rank-r matrices by rank minimization. Furthermore, this value of $m$ precisely matches the dimension of the manifold of all rank-2r matrices. We also prove that for a fixed rank-r matrix, m >= 2nr - r^2 + 1 random measurements are enough to guarantee recovery using rank minimization. These results give a benchmark to which we may compare the efficacy of nuclear-norm minimization

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

Uniqueness conditions for low-rank matrix recovery

Low-rank matrix recovery addresses the problem of recovering an unknown low-rank matrix from few linear measurements. Nuclear-norm minimization is a tractable approach with a recent surge of strong theoretical backing. Analagous to the theory of compressed sensing, these results have required random measurements. For example, m ≥ Cnr Gaussian measurements are sufficient to recover any rank-r n x n matrix with high probability. In this paper we address the theoretical question of how many measurements are needed via any method whatsoever - tractable or not. We show that for a family of random measurement ensembles, m ≥ 4nr-4r^2 measurements are sufficient to guarantee that no rank-2r matrix lies in the null space of the measurement operator with probability one. This is a necessary and sufficient condition to ensure uniform recovery of all rank-r matrices by rank minimization. Furthermore, this value of m precisely matches the dimension of the manifold of all rank-2r matrices. We also prove that for a fixed rank-r matrix, m ≥ 2nr – r^2 + 1 random measurements are enough to guarantee recovery using rank minimization. These results give a benchmark to which we may compare the efficacy of nuclear-norm minimization

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

Efficient and feasible state tomography of quantum many-body systems

We present a novel method to perform quantum state tomography for many-particle systems which are particularly suitable for estimating states in lattice systems such as of ultra-cold atoms in optical lattices. We show that the need for measuring a tomographically complete set of observables can be overcome by letting the state evolve under some suitably chosen random circuits followed by the measurement of a single observable. We generalize known results about the approximation of unitary 2-designs, i.e., certain classes of random unitary matrices, by random quantum circuits and connect our findings to the theory of quantum compressed sensing. We show that for ultra-cold atoms in optical lattices established techniques like optical super-lattices, laser speckles, and time-of-flight measurements are sufficient to perform fully certified, assumption-free tomography. Combining our approach with tensor network methods - in particular the theory of matrix-product states - we identify situations where the effort of reconstruction is even constant in the number of lattice sites, allowing in principle to perform tomography on large-scale systems readily available in present experiments.Comment: 10 pages, 3 figures, minor corrections, discussion added, emphasizing that no single-site addressing is needed at any stage of the scheme when implemented in optical lattice system

¿Qué piensan los actores de la renovación urbana?

Desarrollo Urban

'We looked after people better when we were informal' : the 'quasi-formalisation' of Montevideo's waste-pickers

This article was written thanks to funding from the Economic and Social Research Council (Grant Code: ES/S011048/1).Drawing on participatory research, this article explores the state formalisation of Uruguayan clasificadores (waste‐pickers). It goes beyond the informal/formal binary, instead proposing the concepts of ‘para‐formality’ to describe economic activity that exists in parallel to regulated and taxed spheres, and ‘quasi‐formality’ to describe processes of formalisation that are supported by underlying informal practices. When unregulated, clasificadores enjoyed parallel services in health, finance and social security, implying that benefits of ‘formalisation’ must be explored ethnographically rather than assumed. The persistence of ‘quasi‐formal’ activity within formalised recycling plants complicates simple narratives of informal to formal transitions and suggests that the concept can be useful for the study of labour policies in Latin America and beyondPublisher PDFPeer reviewe

Fondos europeos para la recuperación, transformación y resiliencia : ejecución y auditoría

Disponible en la página web del Observatorio de Contratación Pública.Incluye referencias bibliográficas

Caracterización de la cadena agroproductiva del café en El Salvador

El presente documento de “Caracterización de la Cadena Agroproductiva del Café en El Salvador”, es parte de la ejecución del Plan de Agricultura Familiar para la Seguridad Alimentaria y Nutricional PAF-Cadenas Productivas/Proyecto Rescate y Desarrollo de la Caficultura Nacional, que ejecuta la Cadena Agroproductiva del Café en El Salvador del IICA. Esta caracterización retrata la situación actual en que se encuentra la caficultura y describe los principales retos del sector sus fortalezas, sus debilidades, las amenazas y oportunidades. Se presentan las causas y efectos de los problemas relevantes que el sector enfrenta, pero a la vez la participación de proveedores, productores, beneficiadores y comercializadores le proveen a la caficultura de una fuerza y visión que permite definir acciones para su recuperación. Las actividades sostenidas durante la realización de este estudio facilitaron a representantes del sector, discutieran abiertamente acerca de los problemas que les aquejan y de las posibles soluciones