# Exponential weighted aggregation : oracle inequalities and algorithms

1 Equipe Image - Laboratoire GREYC - UMR6072
GREYC - Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen
Abstract : In many areas of statistics, including signal and image processing, high-dimensional estimation is an important task to recover an object of interest. However, in the overwhelming majority of cases, the recovery problem is ill-posed. Fortunately, even if the ambient dimension of the object to be restored (signal, image, video) is very large, its intrinsic complexity'' is generally small. The introduction of this prior information can be done through two approaches: (i) penalization (very popular) and (ii) aggregation by exponential weighting (EWA). The penalized approach aims at finding an estimator that minimizes a data loss function penalized by a term promoting objects of low (simple) complexity. The EWA combines a family of pre-estimators, each associated with a weight exponentially promoting the same objects of low complexity.This manuscript consists of two parts: a theoretical part and an algorithmic part. In the theoretical part, we first propose the EWA with a new family of priors promoting analysis-group sparse signals whose performance is guaranteed by oracle inequalities. Next, we will analysis the penalized estimator and EWA, with a general prior promoting simple objects, in a unified framework for establishing some theoretical guarantees. Two types of guarantees will be established: (i) prediction oracle inequalities, and (ii) estimation bounds. We will exemplify them for particular cases some of which studied in the literature. In the algorithmic part, we will propose an implementation of these estimators by combining Monte-Carlo simulation (Langevin diffusion process) and proximal splitting algorithms, and show their guarantees of convergence. Several numerical experiments will be considered for illustrating our theoretical guarantees and our algorithms.
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Cited literature [161 references]

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• HAL Id : tel-01690522, version 1

### Citation

Duy Tung Luu. Exponential weighted aggregation : oracle inequalities and algorithms. Complex Variables [math.CV]. Normandie Université, 2017. English. ⟨NNT : 2017NORMC234⟩. ⟨tel-01690522⟩

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