Adaptive Multi-Kernel and Graph Semi-Supervised Support Vector Machine

The performance of semi-supervised support vector machines (\((\mathrm{S^3VMs})\)) depends on the quality of the graph regularizer and the choice of kernel, yet most existing methods keep the graph fixed and tune a single kernel by exhaustive search. We propose AMG-S\((^3)\)VM ( Adaptive Multi-Kernel and Graph \((\mathrm{S^3VM})\) ), which embeds Laplacian refinement and kernel-weight optimization into one unified objective. During each iteration, graph edge weights are updated by a closed-form simplex projection that blends Euclidean proximity with the current margin, while kernel coefficients are adjusted through momentum-augmented mirror descent under simplex constraints. These two updates, followed by a convex \((\mathrm{S^3VM})\) subproblem, yield a block-convex procedure whose objective is proven to decrease monotonically and converge to a critical point; the overhead grows only linearly with the number of base kernels. Extensive experiments on diverse benchmark datasets show that AMG-S\((^3)\)VM consistently surpasses classical Laplacian, deep-kernel, and graph-regularized competitors, yet requires only a coarse grid for a handful of scalar hyper-parameters. The results demonstrate that graph–kernel co-adaptation provides a principled and computationally efficient route to robust semi-supervised learning.