I will explain the problem of portfolio optimization in finance and how sample noise can have catastrophic consequences for risk control. Sample covariance matrices are the result the free product of true covariance and a noise matrix (Wishart or worse if samples are not iid). The inverse problem (retrieving the true covariance) is ill-posed but rotationally invariant estimators can make the best out of insufficient data. This estimator is build using eigenvector overlaps of correlated random matrices. I will discuss similar problems in biophysics/computational biology.
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