Title: Learning Sparse and Low-Rank Priors for Image Recovery via Iterative Reweighted Least Squares Minimization. Abstract: In this work we introduce a novel optimization algorithm for image recovery under learned sparse and low-rank constraints, which are parameterized with weighted extensions of the $\ell_p^p$-vector and $\mathcal{S}_p^p$ Schatten-matrix quasi-norms for $0\!<p\!\le1$, respectively. Our proposed algorithm generalizes the Iteratively Reweighted Least Squares (IRLS) method, used for signal recovery under $\ell_1$ and nuclear-norm constrained minimization. Further, we interpret our overall minimization approach as a recurrent network that we then employ to deal with inverse low-level computer vision problems. Thanks to the convergence guarantees that our IRLS strategy offers, we are able to train the derived reconstruction networks using a memory-efficient implicit back-propagation scheme, which does not pose any restrictions on their effective depth. To assess our networks' performance, we compare them against other existing reconstruction methods on several inverse problems, namely image deblurring, super-resolution, demosaicking and sparse recovery. Our reconstruction results are shown to be very competitive and in many cases outperform those of existing unrolled networks, whose number of parameters is orders of magnitude higher than that of our learned models. 