Sharp oracle inequalities for low-complexity priors

Sharp Oracle Inequalities in Low Rank Estimation

The paper deals with the problem of penalized empirical risk minimization over a convex set of linear functionals on the space of Hermitian matrices with convex loss and nuclear norm penalty. Such penalization is often used in low rank matrix recovery in the cases when the target function can be well approximated by a linear functional generated by a Hermitian matrix of relatively small rank (c...

Sharp Oracle Inequalities for Square Root Regularization

We study a set of regularization methods for high-dimensional linear regression models. These penalized estimators have the square root of the residual sum of squared errors as loss function, and any weakly decomposable norm as penalty function. This fit measure is chosen because of its property that the estimator does not depend on the unknown standard deviation of the noise. On the other hand...

Sharp Oracle Inequalities for Aggregation of Affine Estimators

We consider the problem of combining a (possibly uncountably infinite) set of affine estimators in non-parametric regression model with heteroscedastic Gaussian noise. Focusing on the exponentially weighted aggregate, we prove a PAC-Bayesian type inequality that leads to sharp oracle inequalities in discrete but also in continuous settings. The framework is general enough to cover the combinati...

Aggregation by Exponential Weighting and Sharp Oracle Inequalities

On Sharp Strichartz Inequalities in Low Dimensions

Recently Foschi gave a proof of a sharp Strichartz inequality in one and two dimensions. In this note, a new representation in terms of an orthogonal projection operator is obtained for the space time norm of solutions of the free Schrödinger equation in dimension one and two. As a consequence, the sharp Strichartz inequality follows from the elementary property that orthogonal projections do n...