Graph fractional Laplacian kernel regression for signal prediction

Kernel regression over graphs provides a flexible nonparametric framework for predicting graph signals while allowing the inputs to be agnostic to any graph structure. This paper proposes graph fractional Laplacian kernel regression (FLKR), an extension of the standard graph regularized kernel regression framework that introduces a graph fractional Laplacian into the output smoothness penalty. The resulting model allows continuous control of the spectral weights in the regularizer, thereby adjusting the smoothness of the predicted graph signals. We derive the primal optimality conditions and show that the resulting estimator satisfies a linear matrix normal equation, which admits a closed form characterization and supports efficient numerical solvers. A kernel based implementation is further developed to avoid explicit feature representations and to express training and prediction using only kernel matrices and graph operators. Experiments on both synthetic graph models and real world graph signal datasets demonstrate that appropriately chosen fractional orders can deliver lower prediction error.