I have run two Dantzig Selector solvers after watching the presentation of Vincent Rivoirard on The Dantzig selector for high dimensional statistical problems . I don't know if this is because I was introduced to the subject a second time around but something clicked. I'll have to come back later as it looks to me like some sort of instance of coded aperture imaging. Anyway, I tried both the DS solver of Salman Asif and Justin Romberg at Georgia Tech and that of Emmanuel Candes and Justin Romberg in the L1 magic package. For reasons stemming from the way they are set up I had to change the problem in that in this entry, I am looking at only non complex measurement matrices.
Here are the results for the Donoho-Tanner phase transition with a 0.5% multiplicative noise to the measurement matrices.
Using the Homotopy solver
Clearly, I must not do something well (set the right parameter) with the homotopy solver. It is however very fast and in the noiseless case this is an advantage. At that level of noise, LASSO and the Dantzig Selector seem to provide the same phase transition. Increasing the noise to 1% and using the l1 magic solver we get:
If I am comparing this to the LASSO solver, it looks like the LASSO provides a larger area in the DT phase transition where one can return sparse vectors.
Maybe I should now focus on the beginning of the x-axis, i.e. small number of measurements since my computation have a resolution of 2% (1/50).