Graph signal processing tasks that leverage spectral information typically assume access to the complete graph topology, which is often unavailable in practice. We propose a systematic framework for subgraph filter learning (SFL), where subgraph-supported operators approximate ambient graph filters under partial observations. We formulate SFL as a statistical learning problem in which optimal subgraph operators are inherently data-dependent. To address the difficulty of directly estimating such operators, we develop a subgraph filter algebra based on distance-aware Laplacian constructions, defining a structured and controllable class of filters for effective approximation. We further establish performance risk bounds under the least squares loss, quantifying how well the learned operator approximates the restricted ambient mapping. Experiments real-world datasets show that, for SFL tasks, the proposed algebraic models consistently outperform polynomial filters, distribution-agnostic operators, and direct numerical filter learning baselines that attempt to recover the underlying structure from data.
We aim to learn a sparse and connected graph from sparse data, where the number of observations K can be substantially smaller than the signal dimension N for signals x in R^N, and the underlying distribution is unknown. In this severely ill-posed setting, we incorporate Fiedler number (the second eigenvalue of the graph Laplacian matrix that quantifies connectedness) as a robust regularization term in the sparse graph learning objective. We first develop a greedy algorithm that iteratively selects one edge globally for weakening/removal to reduce the objective, leveraging eigenvalue perturbation theorems that bound the adverse effect of an edge change to the Fiedler number. Next, we design a parallel variant, based on the Cheeger's inequality, that recursively partitions an input graph into two sub-graphs using an approximate Cheeger cut to distributedly find an optimal edge. Simulation experiments show that Fiedler number maximization robustifies sparse graph estimates, outperforming previous sparse graph learning algorithms.
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.
Temporal graph learning is commonly organized around the evolution of node states or the encoding of interaction histories. We study an underexplored, operator-centric question: should the graph propagation mechanism itself evolve over time? We introduce Dynamic Spectral Filtering (DSF), which represents propagation at snapshot t by a Chebyshev polynomial filter with vector-valued, time-dependent coefficients. DSF explicitly treats these compact multi-order coefficients as recurrent temporal states. A recurrent branch proposes updates, while multiplicative global and order-specific gates regulate their magnitude. The temporal state is independent of the number of nodes. On MOOC, Wikipedia, and Reddit temporal link-prediction benchmarks, converged DSF runs attain AP scores of 0.7851, 0.9088, and 0.9860, respectively, with 93K to 133K trainable parameters, 68 to 182 MB peak GPU memory, and 1.6 to 2.1 seconds of training per epoch. Against the closely related DEFT baseline, DSF is better on MOOC, within 0.001 AP on Reddit, and modestly lower on Wikipedia, while using 8.3 to 8.6 times fewer parameters, 25 to 33 times less GPU memory, and 5 to 19 times less time per epoch. Relative to all measured alternatives, it uses 3.3 to 38.6 times less GPU memory. These results support direct spectral-response evolution as a useful temporal inductive bias when computational efficiency is a first-class requirement.