Well-posedness Classes for Sparse Regularization∗

Abstract. Because of their sparsity enhancing properties, `1 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 `1-regularization is equivalent to `0-regularization, which tries to minimise the number of non-zero coefficients. 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 `0-regularization. Moreover, we derive a complete characterisation of those linear operators A for which `0-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 `0-regularization.