用低秩更新建模随时间缓慢变化的图结构，提升时空信号插值效果。
Low-rank Updates in Slowly Time-varying Graphs for Spatial-Temporal Signal Interpolation
- 通过低秩约束建模相邻时刻图结构的变化，保持动态关系的平滑性。
- 联合优化信号与图结构，在有限数据下实现更精准的插值性能。
- 将算法迭代展开为轻量神经网络，适合数据稀缺场景下的参数调优。
图信号处理（GSP）的一个关键假设是存在一个底层图来刻画节点间的成对相似性，从而设计滤波器用于去噪等任务。对于节点间相似性随时间演变的时空数据，静态空间图已不适用。本文针对缓慢时变的成对关系，将两个连续邻接矩阵 $P = W^{(2)} - W^{(1)}$ 之间的变化建模为低秩矩阵。给定初始邻接矩阵 $W^{(1)}$（时间 $t=1$），我们联合插值信号 $x_2$ 并估计 $W^{(2)}$（时间 $t=2$），利用信号平滑性先验和 $P$ 的低秩先验进行交替优化：固定 $W^{(2)}$ 时求解线性系统插值 $x_2$；固定 $x_2$ 时通过近端梯度下降（PGD）更新 $W^{(2)}$。低秩项的近似投影采用快速正交匹配追踪（OMP）算法，在线性时间内完成，该算法从 $W^{(1)}$ 特征向量外积构成的字典 $\mathcal{R}$ 中选取稀疏原子组合。我们将算法迭代展开为多层结构，构建轻量级神经网络以实现有限数据驱动的参数调优。实验表明，所提联合优化方法在信号插值性能上优于现有时变图模型。
原文摘要 · Abstract (English)
A crucial assumption in graph signal processing (GSP) is the existence of an underlying graph that captures the pairwise similarities between nodes, allowing filters to be designed based on this graph for tasks such as denoising. For spatial-temporal data in which node-to-node similarities evolve over time, a static spatial graph is insufficient. In this paper, to represent slowly time-varying pairwise relationships, we model the graph changes in two consecutive adjacency matrices $P = W^{(2)} - W^{(1)}$ across time as a low-rank matrix. % Specifically, given an initial adjacency matrix $W^{(1)}$ at time $t=1$, we jointly interpolate a signal $x_2$ and estimate $W^{(2)}$ at $t=2$ using both a graph signal smoothness prior for $x_2$ and a low-rank prior on $¶$. We alternate optimization steps. With $W^{(2)}$ fixed, $x_2$ is interpolated by solving a linear system. Alternatively, holding $x_2$ fixed, $W^{(2)}$ is updated via proximal gradient descent (PGD). The proximal mapping of the rank term $Gamma(W^{(2)} - W^{(1)})$ is approximated in linear time using a fast orthogonal matching pursuit (OMP) algorithm that selects a sparse combination of atoms from a dictionary $cR$ formed by the outer products of $W^{(1)}$'s eigenvectors. We unroll iterations of our algorithm into layers to build a lightweight neural network for limited data-driven parameter tuning. Experiments show that our joint optimization achieves better signal interpolation compared to existing time-varying graph models.
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