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

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 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 \u3e= 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 \u3e= 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 \u3e= 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

Emergence of chaos in a viscous solution of rods

It is shown that the addition of small amounts of microscopic rods in a viscous fluid at low Reynolds number causes a significant increase of the flow resistance. Numerical simulations of the dynamics of the solution reveal that this phenomenon is associated to a transition from laminar to chaotic flow. Polymer stresses give rise to flow instabilities which, in turn, perturb the alignment of the rods. This coupled dynamics results in the activation of a wide range of scales, which enhances the mixing efficiency of viscous flows.Comment: 4 pages, 5 figures, 1 tabl

Metamorphic titanite–zircon pseudomorphs after igneous zirconolite

The formation of metamorphic zircon after baddeleyite is a well-known reaction that can be used to date the metamorphism of igneous silica-undersaturated rocks. By contrast, metamorphic minerals formed after igneous zirconolite have rarely been reported. In this paper, we document metamorphic titanite + zircon pseudomorphs formed from the metamorphic breakdown of igneous zirconolite in syenodiorite and syenite, in the southeastern Sveconorwegian Province, Sweden. Water-rich fluid influx during tectonometamorphism in epidote–amphibolite-facies metamorphic conditions caused the release of silica during a metamorphic reaction involving igneous feldspar and pyroxene and the simultaneous breakdown of igneous Zr-bearing phases. Typical titanite + zircon intergrowths are elongated or platy titanite crystals speckled with tiny inclusions of zircon. Most intergrowths are smaller than 15 µm; some are subrounded in shape. Locally, bead-like grains of titanite and zircon are intergrown with silicate minerals. The precursor igneous zirconolite was found preserved only in a sample of near-pristine igneous syenodiorite, as remnant grains of mainly &lt; 2 µm in size. Two somewhat larger crystals, 8 and 12 µm, allowed semiquantitative confirmation using microprobe analysis. Analogous with zircon pseudomorphs after baddeleyite, titanite + zircon pseudomorphs after zirconolite potentially offer dating of the metamorphic reaction, although the small size of the crystals makes dating with today's techniques challenging. The scarcity of reports of zirconolite and pseudomorphs reflects that they are either rare or possibly overlooked.</p

Pourquoi les politiques publiques sont-elles si peu suivies d’effets ?:Quelques interrogations

L’insertion des femmes sur le marché du travail a connu à la fois des avancées et des reculs. Si davantage de femmes accèdent à l’éducation supérieure et aux emplois qualifiés, d’autres sont touchées par la précarité et connaissent une dégradation de leurs conditions de travail et de vie. Face à ce constat ambivalent, on peut questionner la mise en œuvre et l’efficacité des politiques qui visent à promouvoir l’égalité entre les femmes et les hommes. Cet article a pour objectif de soulever quelques débats. Le plus souvent, les politiques publiques au sens large (y compris la protection sociale) sont définies en termes de compensation et de correction des inégalités et des discriminations. Mais elles ne concernent pas les causes effectives de l’extension du sous-emploi des femmes, qui relèvent du fonctionnement même du marché du travail. C’est donc la définition des politiques publiques qu’il faut interroger, en dépassant une vision binaire qui oppose d’une part un champ économique extérieur, d’autre part un champ social, juridique et culturel qui, seul, pourrait être l’objet d’inflexions. En réalité, le champ économique est aussi le produit des politiques publiques : la libre-concurrence et la prééminence du marché sont le résultat d’une action volontaire des États. Il faut donc réintégrer les politiques économiques dans le champ de la réflexion sur les moyens de combattre les discriminations à l’encontre des femmes.The integration of women into the labour market has gone through both upswings and downturns. In view of this ambivalent result, we can question the efficiency of public policies set up to overcome gender inequality and fight gender discrimination. Does a real will exist, and if so why is it so inefficient or so poorly implemented? What forms do individual and collective resistance take? Most of the time, public policies are defined in terms of compensation and correction. But they don’t deal with the actual causes of women’s underemployment resulting from labour market adjustments. It is therefore the definition of the public policies that we need to examine, going beyond a binary view that opposes economic issues, on the one hand, to social, juridical and cultural concerns on the other

Sustainable Retailing – Influencing Consumer Behaviour on Food Waste

The aim of this research was to examine the influence of a UK national retailer on its customers' food waste behaviour. Using six communication channels (in‐store magazine, e‐newsletter, Facebook site, product stickers and in‐store demonstrations), Asda presented standard food waste reduction messages to its customers during two time limited periods in 2014 and 2015. Six national surveys over 21 months tracked customers' self‐reported food waste. Our results showed that the combined communication channels and repeated messages over time had a significant effect on reducing food waste of customers. Surprisingly, customers who said they did not recall seeing the messages also reduced their food waste, showing the wider influence of interventions. Those who saw a food waste reduction message saved an estimated £81 annually from reducing food waste. The main conclusion of this paper is that retailers can influence the pro‐environmental behaviour of customers using conventional communication channels; however, repeat messages are needed in order to have a long‐term impact

Overcoming the risk of inaction from emissions uncertainty in smallholder agriculture

The potential for improving productivity and increasing the resilience of smallholder agriculture, while also contributing to climate change mitigation, has recently received considerable political attention (Beddington et al 2012). Financial support for improving smallholder agriculture could come from performance-based funding including sale of carbon credits or certified commodities, payments for ecosystem services, and nationally appropriate mitigation action (NAMA) budgets, as well as more traditional sources of development and environment finance. Monitoring the greenhouse gas fluxes associated with changes to agricultural practice is needed for performance-based mitigation funding, and efforts are underway to develop tools to quantify mitigation achieved and assess trade-offs and synergies between mitigation and other livelihood and environmental priorities (Olander 2012)

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

Remarques sur la relation degré alcoolique-rendement chez la vigne

La relation entre degré alcoolique et rendement a été étudiée chez la vigne. Dans les conditions étudiées, il n'apparaît pas de corrélation.Observations on the relation between alcoholic degree and yield of grapevinesThe relation between alcoholic degree and yield has been studied for the vine. In the conditions of the study, a correlation is not observed between these two factors

A Typology of Child Sponsorship Activity

Framing the debate over child sponsorship in terms of legitimacy and changing perceptions of credible international humanitarian interventions, this chapter takes exception to the tendency of child sponsorship critics to assume that sponsorship funded activity is much the same everywhere and similar today when compared to sponsorship practice in the past. Mindful of ongoing critique of child sponsorship, this chapter seeks to position those international non-governmental organisations that utilise child sponsorship to fund interventions, in a landscape of contested ideas. It argues that informed critique of child sponsorship is best achieved through a typology of funded interventions. Four key types of sponsorship funded activity are identified as emerging over time, some of which are currently deemed to be less legitimate in terms of poverty reduction and are best seen as welfare measures aimed at individual children rather than community development or advocacy activities