Constrained L2-L0 optimization and its application to Single-Molecule

Sparse optimization is crucial in today's society, as this is used in multiple domains, such as denoising, compression, machine learning, and variable selection. Sparse optimization is also vital in single-molecule localization microscopy, a microscopy method widely used in biology. However, obtaining a good sparse solution of a signal is computationally challenging. This thesis focuses on sparse optimization in the form of minimizing the least square loss function under a k-sparse constraint with an L0 pseudo-norm (the constrained L2-L0 problem). We also study the sum of the least square loss function and an L0 penalty term (the penalized L2-L0 problem). Both problems are non-convex, non-continuous, and NP-hard. We propose three new approaches to sparse optimization. We present first a continuous relaxation of the constrained problem and present a method to minimize the proposed relaxation. Secondly, we reformulate the L0 pseudo-norm as a convex minimization problem. This is done by introducing an auxiliary variable, and we present an exact biconvex reformulation of the constrained (CoBic) and penalized (PeBic) problems. Finally, we present a method to minimize the product of the data fidelity term and the regularization term. The latter is still an ongoing research work. We apply the three proposed methods (relaxation, CoBic, and PeBic) to single-molecule localization microscopy and compare them with other commonly used algorithms in sparse optimization. The proposed algorithms' results are as good as the state-of-the-art in grid-based methods. Furthermore, fixing the sparsity constraint constant is usually more intuitive than fixing the penalty parameter, making the constraint approach attractive for applications.