Recovery Algorithms for Vector-Valued Data with Joint Sparsity Constraints

Vector-valued data appearing in concrete applications often possess sparse expansions with respect to a preassigned frame for each vector component individually. Additionally, different components may also exhibit common sparsity patterns. Recently, there were introduced sparsity measures that take into account such joint sparsity patterns, promoting coupling of nonvanishing components. These measures are typically constructed as weighted $\ell_1$ norms of componentwise $\ell_q$ norms of frame coefficients. We show how to compute solutions of linear inverse problems with such joint sparsity regularization constraints by fast thresholded Landweber algorithms. Next we discuss the adaptive choice of suitable weights appearing in the definition of sparsity measures. The weights are interpreted as indicators of the sparsity pattern and are iteratively updated after each new application of the thresholded Landweber algorithm. The resulting two-step algorithm is interpreted as a double-minimization scheme for a suitable target functional. We show its $\ell_2$-norm convergence. An implementable version of the algorithm is also formulated, and its norm convergence is proven. Numerical experiments in color image restoration are presented.