<p>Posterior sampling is explored for high-dimensional linear inverse problems through a transformation of samples drawn from the prior. The approach, which is exact under Gaussian priors and related to ensemble Kalman filtering, is generalized using neural networks. The networks are trained by minimizing the Bayes risk. The generalized transformation results in an improved posterior sampler which scales well to high-dimensional tasks. The approach allows to account for non-Gaussian priors and can be particularly useful when a prior in terms of a generative model is available, representing a database of potential solutions. Statistical properties of the sampling procedure are explored and the procedure is shown to be uniquely determined given its expectation function. The theoretical analysis provided in this work demonstrates that exact posterior sampling is in principle possible. The performance of the procedure is illustrated through the application to synthetic example data and to magnetic resonance imaging data. Software for training and running the deep learning based posterior sampler is made available.</p>