On universal oracle inequalities related to high-dimensional linear models

This paper deals with recovering an unknown vector $\theta$ from the noisy data $Y=A\theta+\sigma\xi$, where $A$ is a known $(m\times n)$-matrix and $\xi$ is a white Gaussian noise. It is assumed that $n$ is large and $A$ may be severely ill-posed. Therefore, in order to estimate $\theta$, a spectral regularization method is used, and our goal is to choose its regularization parameter with the help of the data $Y$. For spectral regularization methods related to the so-called ordered smoothers [see Kneip Ann. Statist. 22 (1994) 835--866], we propose new penalties in the principle of empirical risk minimization. The heuristical idea behind these penalties is related to balancing excess risks. Based on this approach, we derive a sharp oracle inequality controlling the mean square risks of data-driven spectral regularization methods.Comment: Published in at http://dx.doi.org/10.1214/10-AOS803 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Adaptive spectral regularizations of high dimensional linear models

This paper focuses on recovering an unknown vector $\beta$ from the noisy data $Y=X\beta +\sigma\xi$, where $X$ is a known $n\times p$-matrix, $\xi$ is a standard white Gaussian noise, and $\sigma$ is an unknown noise level. In order to estimate $\beta$, a spectral regularization method is used, and our goal is to choose its regularization parameter with the help of the data $Y$. In this paper, we deal solely with regularization methods based on the so-called ordered smoothers and provide some oracle inequalities in the case, where the noise level is unknown

Weak localization in a system with a barrier: Dephasing and weak Coulomb blockade

We non-perturbatively analyze the effect of electron-electron interactions on weak localization (WL) in relatively short metallic conductors with a tunnel barrier. We demonstrate that the main effect of interactions is electron dephasing which persists down to T=0 and yields suppression of WL correction to conductance below its non-interacting value. Our results may account for recent observations of low temperature saturation of the electron decoherence time in quantum dots.Comment: published version, 10 page

Non-local Andreev reflection under ac bias

We theoretically analyze non-local electron transport in multi-terminal normal-metal-superconductor-normal-metal (NSN) devices in the presence of an external ac voltage bias. Our analysis reveals a number of interesting effects, such as, e.g., photon-assisted violation of balance between crossed Andreev reflection (CAR) and elastic cotunneling (EC). We demonstrate that at sufficiently small (typically subgap) frequencies of an external ac signal and at low temperatures the non-local conductance of the NSN device turns negative implying that in this regime CAR contribution to the non-local current dominates over that of EC. Our predictions can be directly tested in future experiments.Comment: published version, 6 pages, 3 figure