Estimating matrices in the symmetric positive-definite (SPD) cone is of interest for many applications ranging from computer vision to graph learning. While there exist various convex optimization-based estimators, they remain limited in expressivity due to their model-based approach. The success of deep learning has thus led many to use neural networks to learn to estimate SPD matrices in a data-driven fashion. For learning structured outputs, one promising strategy involves architectures designed by unrolling iterative algorithms, which potentially benefit from inductive bias properties. However, designing correct unrolled architectures for SPD learning is difficult: they either do not guarantee that their output has all the desired properties, rely on heavy computations, or are overly restrained to specific matrices which hinders their expressivity. In this paper, we propose a novel and generic learning module with guaranteed SPD outputs called SpodNet, that also enables learning a larger class of functions than existing approaches. Notably, it solves the challenging task of learning jointly SPD and sparse matrices. Our experiments demonstrate the versatility of SpodNet layers.

翻译：对称正定（SPD）锥中的矩阵估计在从计算机视觉到图学习的众多应用中具有重要意义。虽然存在各种基于凸优化的估计器，但由于其基于模型的方法，它们在表达能力方面仍然有限。深度学习的成功因此促使许多人使用神经网络以数据驱动的方式学习估计SPD矩阵。对于学习结构化输出，一种有前景的策略涉及通过展开迭代算法设计的架构，这些架构可能受益于归纳偏置特性。然而，为SPD学习设计正确的展开架构是困难的：它们要么不保证其输出具有所有期望的性质，要么依赖于繁重的计算，要么过度局限于特定矩阵从而限制了其表达能力。在本文中，我们提出了一种新颖且通用的学习模块SpodNet，它保证输出SPD矩阵，并且能够学习比现有方法更广泛的函数类别。值得注意的是，它解决了联合学习SPD矩阵和稀疏矩阵这一具有挑战性的任务。我们的实验证明了SpodNet层的多功能性。