On non-asymptotic bounds for estimation in generalized linear models with highly correlated design

We study a high-dimensional generalized linear model and penalized empirical risk minimization with $\ell_1$ penalty. Our aim is to provide a non-trivial illustration that non-asymptotic bounds for the estimator can be obtained without relying on the chaining technique and/or the peeling device.Comment: Published at http://dx.doi.org/10.1214/074921707000000319 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

The Smooth-Lasso and other ℓ1+ℓ2\ell_1+\ell_2-penalized methods

We consider a linear regression problem in a high dimensional setting where the number of covariates $p$ can be much larger than the sample size $n$. In such a situation, one often assumes sparsity of the regression vector, \textit i.e., the regression vector contains many zero components. We propose a Lasso-type estimator $\hat{\beta}^{Quad}$ (where '$Quad$' stands for quadratic) which is based on two penalty terms. The first one is the $\ell_1$ norm of the regression coefficients used to exploit the sparsity of the regression as done by the Lasso estimator, whereas the second is a quadratic penalty term introduced to capture some additional information on the setting of the problem. We detail two special cases: the Elastic-Net $\hat{\beta}^{EN}$, which deals with sparse problems where correlations between variables may exist; and the Smooth-Lasso $\hat{\beta}^{SL}$, which responds to sparse problems where successive regression coefficients are known to vary slowly (in some situations, this can also be interpreted in terms of correlations between successive variables). From a theoretical point of view, we establish variable selection consistency results and show that $\hat{\beta}^{Quad}$ achieves a Sparsity Inequality, \textit i.e., a bound in terms of the number of non-zero components of the 'true' regression vector. These results are provided under a weaker assumption on the Gram matrix than the one used by the Lasso. In some situations this guarantees a significant improvement over the Lasso. Furthermore, a simulation study is conducted and shows that the S-Lasso $\hat{\beta}^{SL}$ performs better than known methods as the Lasso, the Elastic-Net $\hat{\beta}^{EN}$, and the Fused-Lasso with respect to the estimation accuracy. This is especially the case when the regression vector is 'smooth', \textit i.e., when the variations between successive coefficients of the unknown parameter of the regression are small. The study also reveals that the theoretical calibration of the tuning parameters and the one based on 10 fold cross validation imply two S-Lasso solutions with close performance

On the conditions used to prove oracle results for the Lasso

Oracle inequalities and variable selection properties for the Lasso in linear models have been established under a variety of different assumptions on the design matrix. We show in this paper how the different conditions and concepts relate to each other. The restricted eigenvalue condition (Bickel et al., 2009) or the slightly weaker compatibility condition (van de Geer, 2007) are sufficient for oracle results. We argue that both these conditions allow for a fairly general class of design matrices. Hence, optimality of the Lasso for prediction and estimation holds for more general situations than what it appears from coherence (Bunea et al, 2007b,c) or restricted isometry (Candes and Tao, 2005) assumptions.Comment: 33 pages, 1 figur

Relations between some invariants of algebraic varieties in positive characteristic

We discuss relations between certain invariants of varieties in positive characteristic, like the a-number and the height of the Artin-Mazur formal group. We calculate the a-number for Fermat surfacesComment: 13 page

Concept of a laser-plasma based electron source for sub-10 fs electron diffraction

We propose a new concept of an electron source for ultrafast electron diffraction with sub-10~fs temporal resolution. Electrons are generated in a laser-plasma accelerator, able to deliver femtosecond electron bunches at 5 MeV energy with kHz repetition rate. The possibility of producing this electron source is demonstrated using Particle-In-Cell simulations. We then use particle tracking simulations to show that this electron beam can be transported and manipulated in a realistic beamline, in order to reach parameters suitable for electron diffraction. The beamline consists of realistic static magnetic optics and introduces no temporal jitter. We demonstrate numerically that electron bunches with 5~fs duration and containing 1.5~fC per bunch can be produced, with a transverse coherence length exceeding 2~nm, as required for electron diffraction

A New Limit on the Antiproton Lifetime

Measurements of the cosmic ray pbar/p ratio are compared to predictions from an inhomogeneous disk-diffusion model of pbar production and propagation within the Galaxy, combined with a calculation of the modulation of the interstellar cosmic ray spectra as the particles propagate through the heliosphere to the Earth. The predictions agree with the observed pbar/p spectrum. Adding a finite pbar lifetime to the model, we obtain the limit tau_pbar > 0.8 Myr (90 % C.L.).Comment: 13 pages, 3 encapsulated Postscript figures, uses AASTeX; accepted by Astrophysical Journal; minor change

Parallel patterns and trends in functional structures in extinct island mammals

Endemic mammalian species on islands are generally known to have followed a different evolutionary pathway than their mainland relatives. General patterns, such as body size trends, have been described regularly. However, most island mammal species are unique and each of them is adapted to a specific local niche as part of an equally specific ecological assemblage. Therefore, comparing island species across taxa, islands and time is inherently dangerous without understanding the adaptational value of the studied feature in the compared taxa and without taking the ecological setting of the taxa into account. In this contribution, general and recurring patterns are described per taxon. Some features, like body mass change and sturdy limbs, are relatively general, whereas most features, like bone fusions and change of orbital axis, occur only in a very few taxa. Some features are even contradictory, such as brain size and degree of hypsodonty, with each taxon having its own particular design. In conclusion, general patterns are more often than not just trends and need to be applied with caution

Evaluation of linear ozone photochemistry parametrizations in a stratosphere-troposphere data assimilation system

This paper evaluates the performance of various linear ozone photochemistry parametrizations using the stratosphere-troposphere data assimilation system of the Met Office. A set of experiments were run for the period 23 September 2003 to 5 November 2003 using the Cariolle (v1.0 and v2.1), LINOZ and Chem2D-OPP (v0.1 and v2.1) parametrizations. All operational meteorological observations were assimilated, together with ozone retrievals from the Michelson Interferometer for Passive Atmospheric Sounding (MIPAS). Experiments were validated against independent data from the Halogen Occultation Experiment (HALOE) and ozonesondes. Additionally, a simple offline method for comparing the parametrizations is introduced. <br><br> It is shown that in the upper stratosphere and mesosphere, outside the polar night, ozone analyses are controlled by the photochemistry parametrizations and not by the assimilated observations. The most important factor in getting good results at these levels is to pay attention to the ozone and temperature climatologies in the parametrizations. There should be no discrepancies between the climatologies and the assimilated observations or the model, but there is also a competing demand that the climatologies be objectively accurate in themselves. Conversely, in the lower stratosphere outside regions of heterogeneous ozone depletion, the ozone analyses are dominated by observational increments and the photochemistry parametrizations have little influence. <br><br> We investigate a number of known problems in LINOZ and Cariolle v1.0 in more detail than previously, and we find discrepancies in Cariolle v2.1 and Chem2D-OPP v2.1, which are demonstrated to have been removed in the latest available versions (v2.8 and v2.6 respectively). In general, however, all the parametrizations work well through much of the stratosphere, helped by the presence of good quality assimilated MIPAS observations