用可微分编程解决图学习中拉普拉斯伪逆难计算的问题
Difference-of-Convex Regularization for Graph Learning by Differentiable Programming
- 提出基于凸差正则化的图学习框架，绕过直接求伪逆
- 在多种图结构上表现优于传统方法，且稳定收敛
- 适合需要高效图学习的信号处理与机器学习场景
拉普拉斯正则化最小化在信号处理和机器学习中至关重要，但受限于图拉普拉斯伪逆的稠密性和病态性。尽管拉普拉斯矩阵本身稀疏，其伪逆却是稠密且常病态，导致大规模计算不现实。此外，伪逆学习比拉普拉斯学习更困难。为此，本文在给定图拉普拉斯的前提下，提出一种差-凸正则化（DCR）图学习框架，通过正则化最大似然估计近似拉普拉斯伪逆的谱作用，避免直接求逆。通过双变量表示重构拉普拉斯正则非负最小二乘（LR-NNLS），DCR将伪逆学习与实例特定推理解耦，并实现可微分双引导学习下的高效原始解重建。我们建立了DCR算法的稳定性及唯一不动点存在性的理论保证。数值实验表明，该方法在性能上优于凸求解器和图滤波基线，在多种图拓扑下均表现出鲁棒性。
原文摘要 · Abstract (English)
Laplacian-regularized minimization is fundamental in signal processing and machine learning, but is limited by the dense and ill-conditioned nature of the graph Laplacian pseudoinverse. While the Laplacian itself is sparse, its pseudoinverse is dense and often ill-conditioned, rendering direct computation impractical at scale. Moreover, pseudoinverse learning is more challenging than Laplacian learning. To address this challenge, this paper considers the setting where the graph Laplacian is given and proposes a Difference-of-Convex Regularizer (DCR) graph learning framework that approximates the spectral action of the Laplacian pseudoinverse without direct inversion via regularized Maximum Likelihood Estimation (MLE). By reformulating Laplacian-Regularized Nonnegative Least Squares (LR-NNLS) through a dual representation, DCR decouples pseudoinverse learning from instance-specific inference and enables efficient primal solution reconstruction via a differentiable dual-guided learning scheme. We establish theoretical guarantees on stability and the existence of a unique fixed point for DCR algorithm. Numerical experiments demonstrate improved performance over convex solvers and graph filtering baselines and robust performance across diverse graph topologies.
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