Alexandre B. Tsybakov

Simultaneous analysis of Lasso and Dantzig selector

We show that, under a sparsity scenario, the Lasso estimator and the Dantzig selector exhibit similar behavior. For both methods, we derive, in parallel, oracle inequalities for the prediction risk in the general nonparametric regression model, as well as bounds on the l p estimation loss for 1 ≤ p ≤ 2 in the linear model when the number of variables can be much larger than the sample size.

Introduction to Nonparametric Estimation

This is a concise text developed from lecture notes and ready to be used for a course on the graduate level. The main idea is to introduce the fundamental concepts of the theory while maintaining the exposition suitable for a first approach in the field. Therefore, the results are not always given in the most general form but rather under assumptions that lead to shorter or more elegant proofs. The book has three chapters. Chapter 1 presents basic nonparametric regression and density estimators and analyzes their properties. Chapter 2 is devoted to a detailed treatment of minimax lower bounds. Chapter 3 develops more advanced topics: Pinskers theorem, oracle inequalities, Stein shrinkage, and sharp minimax adaptivity. This book will be useful for researchers and grad students interested in theoretical aspects of smoothing techniques. Many important and useful results on optimal and adaptive estimation are provided. As one of the leading mathematical statisticians working in nonparametrics, the author is an authority on the subject.

Sparsity oracle inequalities for the Lasso

This paper studies oracle properties of l1-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 terms of the number of non-zero components of the oracle vec- tor. The results are valid even when the dimension of the model is (much) larger than the sample size and the regression matrix is not positive definite. They can be applied to high-dimensional linear regression, to nonparamet- ric adaptive regression estimation and to the problem of aggregation of arbitrary estimators. AMS 2000 subject classifications: Primary 62G08; secondary 62C20, 62G05, 62G20. Keywords and phrases: sparsity, oracle inequalities, Lasso, penalized least squares, nonparametric regression, dimension reduction, aggregation, mutual coherence, adaptive estimation.

Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion

This paper deals with the trace regression model where

Local Polynomial Estimators of the Volatility Function in Nonparametric Autoregression

n

Optimal rates of aggregation

entries or linear combinations of entries of an unknown

Oracle Inequalities and Optimal Inference under Group Sparsity

m_1\times m_2

Exponential Screening and optimal rates of sparse estimation

matrix

Fast learning rates for plug-in classifiers

A_0

Regularization in statistics

corrupted by noise are observed. We propose a new nuclear norm penalized estimator of