We introduce a novel optimization algorithm for image recovery under learned sparse and low-rank constraints, which we parameterize as 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.

翻译：本文提出一种新的优化算法，用于在学习的稀疏与低秩约束下进行图像恢复。我们将这些约束参数化为$\ell_p^p$向量拟范数和$\mathcal S_p^p$ Schatten-矩阵拟范数的加权扩展形式，其中$0<p\le1$。该算法推广了用于$\ell_1$和核范数约束最小化信号恢复的迭代重加权最小二乘（IRLS）方法。进一步地，我们将整体最小化方法解释为循环网络，并用于处理底层计算机视觉逆问题。得益于IRLS策略的收敛保证，我们能够使用内存高效的隐式反向传播方案训练所导出的重建网络，该方案对其有效深度没有任何限制。为评估网络性能，我们在多个逆问题（即图像去模糊、超分辨率、去马赛克和稀疏恢复）上与其他现有重建方法进行比较。结果表明，我们的重建结果极具竞争力，且在多数情况下优于现有的展开网络，而后者的参数数量比我们的学习模型高出数个数量级。