基于可微分编程的图学习的凸差正则化方法

中文摘要 AI 辅助

拉普拉斯正则化最小化是信号处理与机器学习的基础，但受图拉普拉斯伪逆的稠密性与病态性限制：拉普拉斯本身是稀疏的，其伪逆却稠密且常呈病态，导致大规模场景下直接计算不可行，且伪逆学习比拉普拉斯学习更具挑战性。为解决该问题，本文在给定图拉普拉斯的设定下，提出一种凸差正则化（DCR）图学习框架，通过正则化最大似然估计（MLE）近似拉普拉斯伪逆的谱作用，无需直接求逆。该方法通过对偶表示重构拉普拉斯正则化非负最小二乘（LR-NNLS），将伪逆学习与实例特定推理解耦，可通过可微分对偶引导学习方案高效重构原始解。本文为DCR算法建立了稳定性与唯一不动点存在性的理论保证；数值实验表明，与凸求解器、图滤波基线相比，该方法性能更优，且在不同图拓扑下表现稳健。

英文摘要

Laplacian-regularized minimization is fundamental in signal processing and machine learning, but explicit construction of the dense graph Laplacian pseudoinverse can be computationally expensive. To address this issue, we consider the setting where the graph Laplacian is given and propose a Difference-of-Convex Regularizer (DCR) graph learning framework that learns a reusable approximation of the pseudoinverse action through regularized Maximum Likelihood Estimation (MLE) and shrinkage-regularized Convex--Concave Procedure (CCCP) iterations. For Laplacian-Regularized Nonnegative Least Squares (LR-NNLS), dual and Karush-Kuhn-Tucker (KKT) analysis motivates separating graph-dependent pseudoinverse learning from instance-specific optimization. The learned pseudoinverse is embedded into a differentiable primal optimization procedure as a graph-aware preconditioner for nonnegative reconstruction and can be reused across LR-NNLS instances sharing the same graph. We establish stability and fixed-point existence guarantees for the proposed pseudoinverse-learning iteration. Experiments across different graph topologies and scales demonstrate near-reference accuracy, competitive time-to-moderate-accuracy behavior, and effective cross-instance reuse across independently generated problem instances sharing the same graph.

发表机构

• College of Computing and Data Science, Nanyang Technological University（南洋理工大学计算与数据科学学院）

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