First, somewhat unrelated, here are some Acrobat Reader shortcuts. I’m especially excited about Alt + <– and Alt + –> to go backward or forward a “view”. So for example, in a paper with internal hyperlinks to the citations as they come up, if you click to go to the citations section and check the citation, you can hit “Alt + <–” to go back to the place where you clicked the link.
As for “the right parameter” and some notes on recon algos. I finally understood the “Bregman iteration,” which I would sum up as iterations of solver solutions. So it’s almost like a meta-solver. But taken all together, Bregman iterations wrap around well-known recovery algorithms such as spgl1 and fpc implementations. There’s also orthogonal matching pursuit, gspr, l1-magic, and more. Here are some supporting and related papers from the osher/yin and co. teams:
On the convergence of an active set method for l1 Minimization
relates to fpc_as, the latest iteration of fpc– fpc =fixed point continuation, as = active set
Fast Linearaized Bregman iteration for compressive sensing and sparse denoising
Linearlized seems to mean using the TV norm, but I really don’t know right now.
Bregman iterative algorithms for l1-minimization with applications to compressed sensing
What I would call the main paper for implementing Bregman iterations using any basis pursuit-type problem (BP, BPDN, LASSO, etc) solvers.
Candes, wakin, and Boyd at Stanford cite the above paper, as they provide what looks to be an improvement, or at least a similar but alternate method for updating reconstruction parameters: Enhancing Sparsity by Reweighted l1 minimization (preprint 2009). Also related, and including some phase diagrams to illustrate, we have Maleki and Donoho with Optimally tuned iterative reconstruction algorithms for compressed sensing. The accompanying website devoted to disseminating the results and methods including code, figures, and simulation parametersof that work is here. In the paper they cite the importance of and strive for “promoting reproducible computational research (final paragraph p. 9).” They also point out that the use of phase transitions is much more illuminating and useful than proofs of restricted isometry properties, and much more flexible and accessible for engineers (and scientists!). I totally agree, it’s why I’m using them myself.