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 motivates the use of learning-based approaches to estimate SPD matrices with neural networks in a data-driven fashion. However, designing effective neural architectures for SPD learning is challenging, particularly when the task requires additional structural constraints, such as element-wise sparsity. Current approaches either do not ensure that the output meets all desired properties or lack expressivity. In this paper, we introduce SpodNet, a novel and generic learning module that guarantees SPD outputs and supports additional structural constraints. Notably, it solves the challenging task of learning jointly SPD and sparse matrices. Our experiments illustrate the versatility and relevance of SpodNet layers for such applications.

翻译：在对称正定（SPD）锥中估计矩阵对于从计算机视觉到图学习的众多应用具有重要意义。尽管存在多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受限。深度学习的成功促使人们采用基于学习的方法，以数据驱动的方式通过神经网络估计SPD矩阵。然而，为SPD学习设计有效的神经架构具有挑战性，特别是当任务需要额外的结构约束（例如逐元素稀疏性）时。现有方法要么无法确保输出满足所有期望属性，要么缺乏表达能力。本文提出SpodNet——一种新颖且通用的学习模块，它保证SPD输出并支持额外的结构约束。值得注意的是，该模块解决了联合学习SPD与稀疏矩阵这一具有挑战性的任务。我们的实验展示了SpodNet层在此类应用中的多功能性与相关性。