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Signal Recovery With Certain Involved Convex Data-Fidelity Constraints

This paper proposes an optimization framework that can efficiently deal with convex data-fidelity constraints onto which the metric projections are difficult to compute. Although such an involved data-fidelity constraint is expected to play an important role in signal recovery under non-Gaussian noise contamination, the said difficulty precludes existing algorithms from solving convex optimization problems with the constraint. To resolve this dilemma, we introduce a fixed point set characterization of involved data-fidelity constraints based on a certain computable quasi-nonexpansive mapping. This characterization enables us to mobilize the hybrid steepest descent method to solve convex optimization problems with such a constraint. The proposed framework can handle a variety of involved data-fidelity constraints in a unified manner, without geometric approximation to them. In addition, it requires no computationally expensive procedure such as operator inversion and inner loop. As applications of the proposed framework, we provide image restoration under several types of non-Gaussian noise contamination with illustrative examples.