Elastic-net regularization versus ℓ1-regularization for linear inverse problems with quasi-sparse solutions. (9th December 2016)

• Record Type: Journal Article
• Authors: Chen, De-Han; Hofmann, Bernd; Zou, Jun
• Is Part Of: Inverse problems. Volume 33:Number 1(2017:Jan.)
• Publication Date: 2016-12-09
• Subjects: linear ill-posed problems -- sparsity constraints -- elastic-net regularization -- ℓ1-regularization -- convergence rates -- source conditions; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/33/1/015004
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We consider the ill-posed operator equation Ax = y with an injective and bounded linear operator A mapping between ℓ 2 and a Hilbert space Y, possessing the unique solution x † = { x † k } k = 1 ∞ . For the cases that sparsity x † ∈ ℓ 0 is expected but often slightly violated in practice, we investigate in comparison with the ℓ 1 -regularization the elastic-net regularization, where the penalty is a weighted superposition of the ℓ 1 -norm and the ℓ 2 -norm square, under the assumption that x † ∈ ℓ 1 . There occur two positive parameters in this approach, the weight parameter η and the regularization parameter as the multiplier of the whole penalty in the Tikhonov functional, whereas only one regularization parameter arises in ℓ 1 -regularization. Based on the variational inequality approach for the description of the solution smoothness with respect to the forward operator A and exploiting the method of approximate source conditions, we present some results to estimate the rate of convergence for the elastic-net regularization. The occurring rate function contains the rate of the decay x † k → 0 for k → ∞ and the classical smoothness properties of x † as an element in ℓ 2 .