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On Tuesday, August 8, 2023 at 04:10:26 PM CDT, Daniel Marks <profdc9@...> wrote:


I worked in the field of compressed sensing and I taught a course that included this material. I also wrote several papers on compressed sensing instruments.

I am very familiar with the works of Donoho, Candes, and Tao. The various measures of sparsity, including mutual coherence, the restricted isometry property, L0, and L1 measures.

I also build instruments. I built spectroscopic instruments and inverse scattering radar measurements (which are better conditioned as they are elliptic rather than parabolic systems as in evanescent surface waves) that utilized compressed sensing for data inversion.

I also worked on trying to infer the parameters of distributions of time series with long-memory, long-tailed distribution diffusion processes from measurements of these distributions.

And there's one thing I know: no magic fairy dust turns bad data into good data. You can not wave your sparsity magic wand over data and miraculously get usable data from noise. It doesn't matter if you have the government spend $100 billion to improve a radar signature or oil companies spend $10 billion to find an oil well.

These are the kinds of visions sold by people who want grant money and promise that they can miraculously tease out some data that is somehow latent and overlooked. This is extraordinarily rare as to be unknown.

In the end, you have a physical model for a process. You have possible measurements of that process. You have some inference method, for example, maximum a posteriori. Your estimator can only be good as your model. If your model is well enough behaved, you can get an idea using Fisher information or mininum variance estimation as to the accuracy of the estimator.

I have spend a career solving inverse problems and have been quite successful at this. And I don't promise what I do not think can deliver. And I would not promise that any compressed sensing or estimation would reliably provide an answer, unless there was some reason to believe that the problem was guaranteed by the physical situation to actually satisfy that sparsity constraint.

In reality, most just assume the sparsity constraint, get an answer, and don't bother to compare to reality, or have any sort of cross-validation of the results.

I attach a copy of a lecture for a course that briefly summarizes some basic results in compressed sensing theory as of the time the lecture was written.