Optimal selection of reduced rank estimators of high-dimensional matrices

We introduce a new criterion, the Rank Selection Criterion (RSC), for selecting the optimal reduced rank estimator of the coefficient matrix in multivariate response regression models. The corresponding RSC estimator minimizes the Frobenius norm of the fit plus a regularization term proportional to the number of parameters in the reduced rank model. The rank of the RSC estimator provides a consistent estimator of the rank of the coefficient matrix; in general, the rank of our estimator is a consistent estimate of the effective rank, which we define to be the number of singular values of the target matrix that are appropriately large. The consistency results are valid not only in the classic asymptotic regime, when $n$, the number of responses, and $p$, the number of predictors, stay bounded, and $m$, the number of observations, grows, but also when either, or both, $n$ and $p$ grow, possibly much faster than $m$. We establish minimax optimal bounds on the mean squared errors of our estimators. Our finite sample performance bounds for the RSC estimator show that it achieves the optimal balance between the approximation error and the penalty term. Furthermore, our procedure has very low computational complexity, linear in the number of candidate models, making it particularly appealing for large scale problems. We contrast our estimator with the nuclear norm penalized least squares (NNP) estimator, which has an inherently higher computational complexity than RSC, for multivariate regression models. We show that NNP has estimation properties similar to those of RSC, albeit under stronger conditions. However, it is not as parsimonious as RSC. We offer a simple correction of the NNP estimator which leads to consistent rank estimation.Comment: Published in at http://dx.doi.org/10.1214/11-AOS876 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org) (some typos corrected

Observation of a push force on the end face of a nm fiber taper exerted by outgoing light

There are two different proposals for the momentum of light in a transparent dielectric of refractive index n: Minkowski's version nE/c and Abrahm's version E/(nc), where E and c are the energy and vacuum speed of light, respectively. Despite many tests and debates over nearly a century, momentum of light in a transparent dielectric remains controversial. In this Letter, we report a direct observation of the inward push force on the end face of a free nm fiber taper exerted by the outgoing light. Our results clearly support Abraham momentum. Our experiment also indicates an inward surface pressure on a dielectric exerted by the incident light, different from the commonly recognized pressure due to the specular reflection. Such an inward surface pressure by the incident light may be useful for precise design of the laser-induced inertially-confined fusion.Comment: 9 pages, 3 figures;Accepted for publication as a Letter in Physical Review Letters(CODE: LP11093

Pacifying the Fermi-liquid: battling the devious fermion signs

The fermion sign problem is studied in the path integral formalism. The standard picture of Fermi liquids is first critically analyzed, pointing out some of its rather peculiar properties. The insightful work of Ceperley in constructing fermionic path integrals in terms of constrained world-lines is then reviewed. In this representation, the minus signs associated with Fermi-Dirac statistics are self consistently translated into a geometrical constraint structure (the {\em nodal hypersurface}) acting on an effective bosonic dynamics. As an illustrative example we use this formalism to study 1+1-dimensional systems, where statistics are irrelevant, and hence the sign problem can be circumvented. In this low-dimensional example, the structure of the nodal constraints leads to a lucid picture of the entropic interaction essential to one-dimensional physics. Working with the path integral in momentum space, we then show that the Fermi gas can be understood by analogy to a Mott insulator in a harmonic trap. Going back to real space, we discuss the topological properties of the nodal cells, and suggest a new holographic conjecture relating Fermi liquids in higher dimensions to soft-core bosons in one dimension. We also discuss some possible connections between mixed Bose/Fermi systems and supersymmetry.Comment: 28 pages, 5 figure

Universal statistics of non-linear energy transfer in turbulent models

A class of shell models for turbulent energy transfer at varying the inter-shell separation, $\lambda$, is investigated. Intermittent corrections in the continuous limit of infinitely close shells ($\lambda \rightarrow 1$) have been measured. Although the model becomes, in this limit, non-intermittent, we found universal aspects of the velocity statistics which can be interpreted in the framework of log-poisson distributions, as proposed by She and Waymire (1995, Phys. Rev. Lett. 74, 262). We suggest that non-universal aspects of intermittency can be adsorbed in the parameters describing statistics and properties of the most singular structure. On the other hand, universal aspects can be found by looking at corrections to the monofractal scaling of the most singular structure. Connections with similar results reported in other shell models investigations and in real turbulent flows are discussed.Comment: 4 pages, 2 figures available upon request to [email protected]

Predicting Allograft Requirement in the Management of Patients With Major Burn Injuries

• Early debridement and coverage of burn wounds saves lives. • Allograft is the ‘gold-standard’ for temporary coverage of acute burns

Accelerating universe emergent from the landscape

We propose that the existence of the string landscape suggests the universe can be in a quantum glass state, where an extremely large viscosity is generated, and long distance dynamics slows down. At the same time, the short distance dynamics is not altered due to the separation of time scales. This scenario can help to understand some controversies in cosmology, for example the natural existence of slow roll inflation and dark energy in the landscape, the apparent smallness of the cosmological constant. We see also that moduli stabilization is no longer necessary. We further identify the glass transition point, where the viscosity diverges, as the location of the cosmic horizon. We try to reconstruct the geometry of the accelerating universe from the structure of the landscape, and find that the metric should have an infinite jump when crossing the horizon. We predict that the static coordinate metric for dS space breaks down outside the horizon.Comment: 20 pages, no figures, harvma