Standard regularization methods that are used to compute solutions to ill-posed inverse problems require knowledge of the forward model. In many real-life applications, the forward model is not known, but training data is readily available. In this paper, we develop a new framework that uses training data, as a substitute for knowledge of the forward model, to compute an optimal low-rank regularized inverse matrix directly, allowing for very fast computation of a regularized solution. We consider a statistical framework based on Bayes and empirical Bayes risk minimization to analyze theoretical properties of the problem. We propose an efficient rank update approach for computing an optimal low-rank regularized inverse matrix for various error measures. Numerical experiments demonstrate the benefits and potential applications of our approach to problems in signal and image processing.