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We are interested in the problem of estimating a regression function  φ  observed with a correlated noise  Y  =  φ ( X )+ U . Contrary to the usual regression model,  U  is not centered conditionaly on  X  but rather on an observed variable  W . Hence this model turns to be a difficult inverse problem where the corresponding operator is unknown since it is related to the joint distribution of ( X,W ). We focus on the case where the eigenvalues of the corresponding operator are observed with small perturbations and, using a well adapted spectral cut-off estimation procedure, we build a data driven estimates and derive an oracle inequality.

Adaptive estimation for an inverse regression model with unknown operator