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I am looking for new ideas how to construct a guess for a (positive, hermitian) matrix A given some matrix-vector products Ax (with random vectors x). One such method would be to perform rank one updates to A each time a new Ax becomes available, and this is what is used in for example the BFGS update formula in quasi-Newton methods. However, this just updates a one dimensional subspace of the matrix. In my application it would be more natural to scale the whole matrix instead of performing a rank one update, but just a simple scaling is not enough because it cannot be consistent for all the Ax's. I have a rather good approximation to A to start with, so I know roughly the distributions of eigenvalues if that can help. My biggest problem is that I don't know how to even define the problem properly..

Perhaps I can sharpen the question with some feelback.

Edit: Learned about shrinkage estimation, perhaps that is the way to go.

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    Can you give some more details. You say you have some information about $A$, that might be expressed as a prior distribution in a bayes analysis, and the random matrix-vector products would be the data used to update the prior to give the posterior. Or maybe methods from tomography?2014-04-09

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