Related papers: Convergence rates for Penalised Least Squares Esti…
This paper deals with the nonparametric estimation in heteroscedastic regression $ Y_i=f(X_i)+\xi_i, \: i=1,...,n $, with incomplete information, i.e. each real random variable $ \xi_i $ has a density $ g_{i} $ which is unknown to the…
We consider the stable approximation of sparse solutions to non-linear operator equations by means of Tikhonov regularization with a subquadratic penalty term. Imposing certain assumptions, which for a linear operator are equivalent to the…
In this manuscript, we study quantile regression in partial functional linear model where response is scalar and predictors include both scalars and multiple functions. Wavelet basis are adopted to better approximate functional slopes while…
We consider both $\ell _{0}$-penalized and $\ell _{0}$-constrained quantile regression estimators. For the $\ell _{0}$-penalized estimator, we derive an exponential inequality on the tail probability of excess quantile prediction risk and…
A numerical analysis for the fully discrete approximation of an operator Lyapunov equation related to linear SPDEs (stochastic partial differential equations) driven by multiplicative noise is considered. The discretization of the Lyapunov…
Bayesian approach to inverse problems is studied in the case where the forward map is a linear hypoelliptic pseudodifferential operator and measurement error is additive white Gaussian noise. The measurement model for an unknown Gaussian…
We provide a priori error estimates for variational approximations of the ground state eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form $-{div} (A\nabla u) + Vu + f(u^2) u = \lambda u$, $\|u\|_{L^2}=1$. We…
This paper investigates tradeoffs among optimization errors, statistical rates of convergence and the effect of heavy-tailed errors for high-dimensional robust regression with nonconvex regularization. When the additive errors in linear…
A common way to estimate an unknown convex regression function $f_0: \Omega \subset \mathbb{R}^d \rightarrow \mathbb{R}$ from a set of $n$ noisy observations is to fit a convex function that minimizes the sum of squared errors. However,…
We study the problem of estimating the coefficients in linear ordinary differential equations (ODE's) with a diverging number of variables when the solutions are observed with noise. The solution trajectories are first smoothed with local…
This paper considers the penalized least squares estimator with arbitrary convex penalty. When the observation noise is Gaussian, we show that the prediction error is a subgaussian random variable concentrated around its median. We apply…
This paper is about learning the parameter-to-solution map for systems of partial differential equations (PDEs) that depend on a potentially large number of parameters covering all PDE types for which a stable variational formulation (SVF)…
The problem of estimating the shift (or, equivalently, the center of symmetry) of an unknown symmetric and periodic function $f$ observed in Gaussian white noise is considered. Using the blockwise Stein method, a penalized profile…
We consider a $l_1$-penalization procedure in the non-parametric Gaussian regression model. In many concrete examples, the dimension $d$ of the input variable $X$ is very large (sometimes depending on the number of observations). Estimation…
Since its early use in least squares regression problems, the l1-penalization framework for variable selection has been employed in conjunction with a wide range of loss functions encompassing regression, classification and survival…
Motivated by robust and quantile regression problems, we investigate the stochastic gradient descent (SGD) algorithm for minimizing an objective function $f$ that is locally strongly convex with a sub--quadratic tail. This setting covers…
This paper studies oracle properties of $\ell_1$-penalized least squares in nonparametric regression setting with random design. We show that the penalized least squares estimator satisfies sparsity oracle inequalities, i.e., bounds in…
A new method is proposed for variable screening, variable selection and prediction in linear regression problems where the number of predictors can be much larger than the number of observations. The method involves minimizing a penalized…
In this paper, we consider the nonparametric regression problem with multivariate predictors. We provide a characterization of the degrees of freedom and divergence for estimators of the unknown regression function, which are obtained as…
Regularization is widely used in statistics and machine learning to prevent overfitting and gear solution towards prior information. In general, a regularized estimation problem minimizes the sum of a loss function and a penalty term. The…