Well-posedness classes for sparse regularization

Because of their sparsity enhancing properties, ℓ¹ penalty terms have recently received much attention in the field of inverse problems. Also, it has been shown that certain properties of the linear operator A to be inverted imply that ℓ¹-regularization is equivalent to ℓ°-regularization, which tries to minimise the number of non-zero coeffcients. In the context of compressed sensing, one usually assumes a restricted isometry property, which requires that the operator A acts almost like an isometry on certain low dimensional sub-spaces. In this paper, we show that similar properties appear naturally when one studies the question of well-posedness of ℓ°-regularization. Moreover, we derive a complete characterisation of those linear operators A for which ℓ°-regularization is wellposed. It turns out that neither boundedness nor invertibility of A are necessary conditions; compact operators, however, are shown not to be suited for ℓ°-regularization.

Keywords

sparsity, quasi-solutions, restricted isometry

2010 Mathematics Subject Classification

47A52, 65J20

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Published 29 July 2011