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For random matrix models, the parameter estimation based on the traditionallikelihood functions is not straightforward in particular when we have only onesample matrix. We introduce a new parameter optimization method for randommatrix models which works even in such a case. The method is based on thespectral distribution instead of the traditional likelihood. In the method, theCauchy noise has an essential role because the free deterministic equivalent,which is a tool in free probability theory, allows us to approximate thespectral distribution perturbed by Cauchy noises by a smooth and accessibledensity function.  Moreover, we study an asymptotic property of determination gap, which has asimilar role as generalization gap. Besides, we propose a new dimensionalityrecovery method for the signal-plus-noise model, and experimentally demonstratethat it recovers the rank of the signal part even if the true rank is notsmall. It is a simultaneous rank selection and parameter estimation procedure.

Cauchy noise loss for stochastic optimization of random matrix models via free deterministic equivalents