I am currently interested in random matrix theory and application. For the application, we know that the phase retrieval/matrix completion/compressed sensing achieved great success with the help of the random matrix theory, or so-called the randomness inside. But I do know why the randomness would help reconstruction? For example, I could use deterministic measurement matrix such as Hadamard matrix, instead of Gaussian random variable. From the perspective of the optimization algorithm (reconstruction algorithm), the measurement matrix provides the prior information hence algorithm would converge to the local/global optimal solution. Random/non-random measurement procedures should be the same.
Is there any reference that provides the theoretical/empirical analysis of the superior of random measurement? Thanks in advance!
For compressive sensing, the restricted isometry property (RIP) is really important. For random matrices (Gaussian random matrices), this property is easy to satisfied. But for deterministic measurement matrix, I don't think so.
Another thing, we could give optimal bound on the number of measurement m by using random matrices. But we fail to do it by using deterministic measurement matrix.
I think constructing deterministic measurement matrix for compressive sensing is still unsolved and lots of people are trying to construct such matrices which are provably optimal in compressive sensing. It is an open problem and good research area.