Statistical Science

Sparse Estimation by Exponential Weighting

Philippe Rigollet and Alexandre B. Tsybakov

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Abstract

Consider a regression model with fixed design and Gaussian noise where the regression function can potentially be well approximated by a function that admits a sparse representation in a given dictionary. This paper resorts to exponential weights to exploit this underlying sparsity by implementing the principle of sparsity pattern aggregation. This model selection take on sparse estimation allows us to derive sparsity oracle inequalities in several popular frameworks, including ordinary sparsity, fused sparsity and group sparsity. One striking aspect of these theoretical results is that they hold under no condition in the dictionary. Moreover, we describe an efficient implementation of the sparsity pattern aggregation principle that compares favorably to state-of-the-art procedures on some basic numerical examples.

Article information

Source
Statist. Sci., Volume 27, Number 4 (2012), 558-575.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ss/1356098556

Digital Object Identifier
doi:10.1214/12-STS393

Mathematical Reviews number (MathSciNet)
MR3025134

Zentralblatt MATH identifier
1331.62351

Keywords
High-dimensional regression exponential weights sparsity fused sparsity group sparsity sparsity oracle inequalities sparsity pattern aggregation sparsity prior sparse regression

Citation

Rigollet, Philippe; Tsybakov, Alexandre B. Sparse Estimation by Exponential Weighting. Statist. Sci. 27 (2012), no. 4, 558--575. doi:10.1214/12-STS393. https://projecteuclid.org/euclid.ss/1356098556


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