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This paper introduces a Difference-of-Convex Regularizer (DCR) framework for graph learning that effectively approximates the spectral action of the graph Laplacian pseudoinverse without direct inversion, leveraging regularized Maximum Likelihood Estimation (MLE). By reformulating Laplacian-Regularized Nonnegative Least Squares (LR-NNLS) through a dual representation, the method decouples pseudoinverse learning from instance-specific inference, allowing for efficient solution reconstruction. Theoretical guarantees on stability and unique fixed points are established, and numerical experiments show that DCR outperforms existing convex solvers and graph filtering methods across various graph topologies.
DCR achieves robust graph learning performance by sidestepping the computational pitfalls of Laplacian pseudoinverse inversion.
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.
