Dirk A. Lorenz

Iterated Hard Shrinkage for Minimization Problems with Sparsity Constraints

A new iterative algorithm for the solution of minimization problems in infinite-dimensional Hilbert spaces which involve sparsity constraints in form of

Convergence rates and source conditions for Tikhonov regularization with sparsity constraints

\ell^{p}

Regularization with non-convex separable constraints

-penalties is proposed. In contrast to the well-known algorithm considered by Daubechies, Defrise, and De Mol, it uses hard instead of soft shrinkage. It is shown that the hard shrinkage algorithm is a special case of the generalized conditional gradient method. Convergence properties of the generalized conditional gradient method with quadratic discrepancy term are analyzed. This leads to strong convergence of the iterates with convergence rates

Inline hologram reconstruction with sparsity constraints

\mathcal{O}(n^{-1/2})

Greedy solution of ill-posed problems: error bounds and exact inversion

and

A semismooth Newton method for Tikhonov functionals with sparsity constraints

\mathcal{O}(\lambda^n)

An Inertial Forward-Backward Algorithm for Monotone Inclusions

for

A generalized conditional gradient method for nonlinear operator equations with sparsity constraints

p=1

On Conditions for Convergence to Consensus

and

Minimization of Non-smooth, Non-convex Functionals by Iterative Thresholding

1 < p \leq 2