AI 中文总结
本研究针对从含噪矢量值信号中学习一致连接图的逆问题，提出结构化连接图学习（SCGL）算法，可联合估计去噪信号、图拓扑与局部参考系，提升拓扑几何恢复及去噪压缩性能。
AI 中文摘要
连接图（CGs）是经典图的扩展，其节点关联矢量值信号，边关联正交传输映射，是同步和基于流形的信号处理的自然模型。尽管其应用日益广泛，但从观测中直接学习CG仍具挑战性，因为网络拓扑与底层几何结构通过非欧几里得正交约束耦合。本研究通过从含噪矢量值信号中学习一致连接图来解决这一逆问题。利用一致CG的谱特性，我们构建了结构化学习问题，联合估计去噪信号、图拓扑和节点级局部参考系。该公式将学习到的连接拉普拉斯谱与底层组合拉普拉斯谱耦合，在保证非平凡全局截面空间的同时，引入显式谱和拓扑先验。我们开发了结构化连接图学习（SCGL），这是一种块坐标算法，结合闭式更新、流形投影和谱约束，可收敛到所得非凸问题的驻点。数值实验表明，SCGL在拓扑和几何恢复上优于对比方法，同时能生成有效的去噪和信号压缩基。
英文摘要
Connection graphs (CGs) extend classical graphs by associating vector-valued signals to nodes and orthogonal transport maps across edges, making them a natural model for synchronization and manifold-based signal processing. Despite their growing use, learning CGs directly from observations remains challenging because the network topology and the underlying geometric structure are coupled through non-Euclidean orthogonality constraints. In this work, we address this inverse problem by learning a consistent connection graph from noisy vector-valued signals. Exploiting the spectral characterization of consistent CGs, we formulate a structured learning problem that jointly estimates a denoised signal, the graph topology, and node-wise local reference frames. The proposed formulation couples the spectrum of the learned connection Laplacian to that of an underlying combinatorial Laplacian, enabling explicit spectral and topological priors while guaranteeing a nontrivial global-section space. We develop Structured Connection Graph Learning (SCGL), a block-coordinate algorithm that combines closed-form updates, manifold projections, and spectral constraints, and converges to stationary points of the resulting nonconvex problem. Numerical experiments show that SCGL improves topology and geometry recovery over competing approaches, while also yielding effective denoising and signal-compression bases.
