Matrix-free interior point method for compressed sensing problems

We consider a class of optimization problems for sparse signal reconstruction which arise in the field of Compressed Sensing (CS). A plethora of approaches and solvers exist for such problems, for example GPSR, FPC AS, SPGL1, NestA, `1 `s, PDCO to mention a few. CS applications lead to very well conditioned optimization problems and therefore can be solved easily by simple first-order methods. Interior point methods (IPMs) rely on the Newton method hence they use the second-order information. They have numerous advantageous features and one clear drawback: being the second-order approach they need to solve linear equations and this operation has (in the general dense case) an O(n) computational complexity. Attempts have been made to specialize IPMs to sparse reconstruction problems and they have led to interesting developments implemented in `1 `s and PDCO softwares. We go a few steps further. First, we use the matrixfree IPM, an approach which redesigns IPM to avoid the need to explicitly formulate (and store) the Newton equation systems. Secondly, we exploit the ? Supported by EPSRC Grant EP/I017127/1 Kimon Fountoulakis School of Mathematics and Maxwell Institute, The University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom. E-mail: [email protected] Tel.: +44 131 650 5083, Fax: +44 131 650 6553 Jacek Gondzio School of Mathematics and Maxwell Institute, The University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom. E-mail: [email protected] Tel.: +44 131 650 8574, Fax: +44 131 650 6553 Pavel Zhlobich School of Mathematics and Maxwell Institute, The University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom. E-mail: [email protected] Tel.: +44 131 650 5044, Fax: +44 131 650 6553