The Expected Patch Log-Likelihood algorithm (EPLL) and its extensions have shown good performances for image denoising. It estimates a Gaussian mixture model (GMM) from a training database of image patches and it uses the GMM as a prior for denoising. In this work, we adapt the sketching framework to carry out the compressive estimation of Gaussian mixture models with low rank covariances for image patches. With this method, we estimate models from a compressive representation of the training data with a learning cost that does not depend on the number of items in the database. Our method adds another dimension reduction technique (low-rank modeling of covariances) to the existing sketching methods in order to reduce the dimension of model parameters and to add flexibility to the modeling. We test our model on synthetic data and real large-scale data for patch-based image denoising. We show that we can produce denoising performance close to the models estimated from the original training database, opening the way for the study of denoising strategies using huge patch databases.