On the convergence of iterative shrinkage algorithms with adaptive discrepancy terms

In this paper, the inversion of a linear operator is tackled by a procedure called iterative shrinkage. Iterative shrinkage is a procedure that minimizes a functional balancing quadratic discrepancy terms with Lp regularization terms. In this work, we propose to replace the classical quadratic discrepancy terms with adaptive ones. These adaptive terms rely on adapted projections on a suitable basis. Two versions of these adaptive terms are proposed (one with a straightforward use of the projections and the other with relaxed projections) together with iterative algorithms minimizing the obtained functional. We prove the convergence and stability of corresponding algorithms. Moreover we prove that for a straightforward use of these adaptive projections, although the process is consistent, valuable information may be lost, which is not the case with the ``relaxed'' projections. We illustrate both algorithms on multispectral astronomical data.