Difference-of-Convex Regularization for Graph Learning by Differentiable Programming

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.

Accelerating Regularized Attention Kernel Regression for Spectrum Cartography

Spectrum cartography reconstructs spatial radio fields from sparse and heterogeneous wireless measurements, underpinning many sensing and optimization tasks in wireless networks. Attention mechanisms have recently enabled adaptive measurement aggregation via attention kernel-based formulations. However, the resulting exponential kernels exhibit severe spectral imbalance, inducing large condition numbers that render standard iterative solvers ineffective for regularized attention kernel regression. This paper proposes a Learning-based Attention Kernel Regression (LAKER) algorithm for accelerating regularized attention kernel regression in spectrum cartography. The key idea is to learn a data-dependent preconditioner that captures the inverse spectral structure of the attention kernel system, directly reducing the condition number bottleneck. The preconditioner is obtained by solving a regularized maximum-likelihood estimation problem via a shrinkage-regularized convex--concave procedure, and is integrated with a preconditioned conjugate gradient solver for efficient optimization, whose solution is used for radio map reconstruction. Extensive experiments demonstrate that LAKER significantly reduces condition numbers by up to three orders of magnitude, accelerates convergence by over twenty-fold compared to baselines, and maintains high reconstruction accuracy, establishing learning-based preconditioning as an effective approach for attention kernel regression in spectrum cartography.

Learning to Optimize by Differentiable Programming

Solving massive-scale optimization problems requires scalable first-order methods with low per-iteration cost. This tutorial highlights a shift in optimization: using differentiable programming not only to execute algorithms but to learn how to design them. Modern frameworks such as PyTorch, TensorFlow, and JAX enable this paradigm through efficient automatic differentiation. Embedding first-order methods within these systems allows end-to-end training that improves convergence and solution quality. Guided by Fenchel-Rockafellar duality, the tutorial demonstrates how duality-informed iterative schemes such as ADMM and PDHG can be learned and adapted. Case studies across LP, OPF, Laplacian regularization, and neural network verification illustrate these gains.