Low rank matrix recovery from rank one measurements

We study the recovery of Hermitian low rank matrices $X \in \mathbb{C}^{n \times n}$ from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form $a_j a_j^*$ for some measurement vectors $a_1,...,a_m$, i.e., the measurements are given by $y_j = \mathrm{tr}(X a_j a_j^*)$. The case where the matrix $X=x x^*$ to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements, $y_j = |\langle x,a_j\rangle|^2$ via the PhaseLift approach, which has been introduced recently. We derive bounds for the number $m$ of measurements that guarantee successful uniform recovery of Hermitian rank $r$ matrices, either for the vectors $a_j$, $j=1,...,m$, being chosen independently at random according to a standard Gaussian distribution, or $a_j$ being sampled independently from an (approximate) complex projective $t$-design with $t=4$. In the Gaussian case, we require $m \geq C r n$ measurements, while in the case of $4$-designs we need $m \geq Cr n \log(n)$. Our results are uniform in the sense that one random choice of the measurement vectors $a_j$ guarantees recovery of all rank $r$-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate $4$-designs generalizes and improves a recent bound on phase retrieval due to Gross, Kueng and Krahmer. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.Comment: 24 page

Election outcomes and maximizing turnout: Modelling the effect

An election outcome reflects institutional, behavioural and attitudinal influences. We set out a model showing it is a function of the electoral system, the offices at stake and the number of parties competing as well as the choices of voters and the level of turnout. Therefore, any attempt to estimate the impact of increased turnout on an election outcome must go beyond a comparison of the party preferences of voters and non-voters. This paper presents a model which integrates six different types of influences that collectively determine election outcomes. It demonstrates empirically that maximum turnout falls well short of 100 percent turnout. It also shows the effect of proportional representation and multiple parties in reducing the net benefit that any one party could expect from increased turnout and the inadequacy of using shares of the popular vote to predict increased turnout effects in the United States. It leaves open the normative debate between advocates of civic participation and the libertarian value of being free not to vote. --

Marriage Matters: Spousal Similarity in Life Satisfaction

Examined the concurrent and cross-lagged spousal similarity in life satisfaction over a 21-year period. Analyses were based on married couples (N = 847) in the German Socio-Economic Panel (SOEP). Concurrent spousal similarity was considerably higher than one-year retest similarity, revealing spousal similarity in the variable component of life satisfac-tion. Spousal similarity systematically decreased with length of retest interval, revealing simi-larity in the changing component of life satisfaction. Finally, there was considerable spousal similarity in the stable component of life satisfaction over 20-years. The implications of these findings for causal theories of life satisfaction and studies in line with behavioural genetics are discussedSubjective Well Being, Life Satisfaction, Marriage, Couples, Spousal Similarity, Heritability, Assortative Mating, Longitudinal Panel, SOEP

Stable low-rank matrix recovery via null space properties

The problem of recovering a matrix of low rank from an incomplete and possibly noisy set of linear measurements arises in a number of areas. In order to derive rigorous recovery results, the measurement map is usually modeled probabilistically. We derive sufficient conditions on the minimal amount of measurements ensuring recovery via convex optimization. We establish our results via certain properties of the null space of the measurement map. In the setting where the measurements are realized as Frobenius inner products with independent standard Gaussian random matrices we show that $10 r (n_1 + n_2)$ measurements are enough to uniformly and stably recover an $n_1 \times n_2$ matrix of rank at most $r$. We then significantly generalize this result by only requiring independent mean-zero, variance one entries with four finite moments at the cost of replacing $10$ by some universal constant. We also study the case of recovering Hermitian rank-$r$ matrices from measurement matrices proportional to rank-one projectors. For $m \geq C r n$ rank-one projective measurements onto independent standard Gaussian vectors, we show that nuclear norm minimization uniformly and stably reconstructs Hermitian rank-$r$ matrices with high probability. Next, we partially de-randomize this by establishing an analogous statement for projectors onto independent elements of a complex projective 4-designs at the cost of a slightly higher sampling rate $m \geq C rn \log n$. Moreover, if the Hermitian matrix to be recovered is known to be positive semidefinite, then we show that the nuclear norm minimization approach may be replaced by minimizing the $\ell_2$-norm of the residual subject to the positive semidefinite constraint. Then no estimate of the noise level is required a priori. We discuss applications in quantum physics and the phase retrieval problem.Comment: 26 page

A large neighbourhood based heuristic for two-echelon routing problems

In this paper, we address two optimisation problems arising in the context of city logistics and two-level transportation systems. The two-echelon vehicle routing problem and the two-echelon location routing problem seek to produce vehicle itineraries to deliver goods to customers, with transits through intermediate facilities. To efficiently solve these problems, we propose a hybrid metaheuristic which combines enumerative local searches with destroy-and-repair principles, as well as some tailored operators to optimise the selections of intermediate facilities. We conduct extensive computational experiments to investigate the contribution of these operators to the search performance, and measure the performance of the method on both problem classes. The proposed algorithm finds the current best known solutions, or better ones, for 95% of the two-echelon vehicle routing problem benchmark instances. Overall, for both problems, it achieves high-quality solutions within short computing times. Finally, for future reference, we resolve inconsistencies between different versions of benchmark instances, document their differences, and provide them all online in a unified format

Resonant-plane locking and spin alignment in stellar-mass black-hole binaries: a diagnostic of compact-binary formation

We study the influence of astrophysical formation scenarios on the precessional dynamics of spinning black-hole binaries by the time they enter the observational window of second- and third-generation gravitational-wave detectors, such as Advanced LIGO/Virgo, LIGO-India, KAGRA and the Einstein Telescope. Under the plausible assumption that tidal interactions are efficient at aligning the spins of few-solar mass black-hole progenitors with the orbital angular momentum, we find that black-hole spins should be expected to preferentially lie in a plane when they become detectable by gravitational-wave interferometers. This "resonant plane" is identified by the conditions \Delta\Phi=0{\deg} or \Delta\Phi=+/-180{\deg}, where \Delta\Phi is the angle between the components of the black-hole spins in the plane orthogonal to the orbital angular momentum. If the angles \Delta \Phi can be accurately measured for a large sample of gravitational-wave detections, their distribution will constrain models of compact binary formation. In particular, it will tell us whether tidal interactions are efficient and whether a mechanism such as mass transfer, stellar winds, or supernovae can induce a mass-ratio reversal (so that the heavier black hole is produced by the initially lighter stellar progenitor). Therefore our model offers a concrete observational link between gravitational-wave measurements and astrophysics. We also hope that it will stimulate further studies of precessional dynamics, gravitational-wave template placement and parameter estimation for binaries locked in the resonant plane.Comment: 26 pages, 11 figures, 3 tables, accepted in Physical Review D. 4 movies illustrating resonance locking are available online: for links, see footnote 8 of the pape

CONSUMER SHOPPING PATTERNS AND PREFERENCES FOR PACKAGE SIZES

Consumer/Household Economics,

Different functions, different histories. Modal particles and discourse markers from a diachronic point of view

One of the main concerns of recent research in discourse markers, modal particles and related elements has been the problem of a neat categorical delimitation between the major classes involved. Comparing the Spanish discourse particle bien and its French cognate modal particle bien, we show that the functional difference between discourse markers and modal particles can be accounted for in diachronic terms. In other words, discourse markers and modal particles arise in different diachronic pathways, and ultimately from different pragmatic strategies. Strategies which eventually yield discourse markers are related to the joint coordination of human interaction. In contrast, strategies which give rise to modal particles make reference to the status of a given proposition for the ensuing discourse. Our results suggest that there are levels of generalization on semantic change below the overarching tendencies of subjectification (Traugott and Dasher 2002). More importantly, we provide specific motivations for these levels of generalization, and ultimately for subjectification itself

Micellar dye shuttle between water and an ionic liquid

The reversible transfer of poly(2-ethyl-2-oxazoline)-block-poly(2-nonyl-2-oxazoline) nanocarriers comprising encapsulated dyes is demonstrated between water and an ionic liquid. This dye transfer concept is shown to be applicable for loading and delivery of dye molecules as well as to provide a protective environment for the encapsulated dye