Random Wavelet Features for Graph Kernel Machines
Valentin de Bassompierre, Jean-Charles Delvenne, Laurent Jacques
TL;DR
This work tackles scalable graph kernel computation by introducing randomized spectral node embeddings that approximate a Laplacian-based kernel $\boldsymbol\Gamma = h(\boldsymbol L)$ with a rank-$K$ matrix $\tilde{\boldsymbol\Gamma} = \boldsymbol\Phi^\top \boldsymbol\Phi$. The method proceeds in two stages: range finding via polynomial-filtered random signals to approximate the top-$K$ spectral subspace, and kernel embedding yielding $\boldsymbol\Phi = (h^{1/2}(\boldsymbol L)\boldsymbol Q)^\top$ so that $\tilde{\boldsymbol\Gamma}$ captures the dominant kernel components. The paper provides theoretical error bounds, connects to Randomized SVD, analyzes computational complexity, and demonstrates empirically that the approach outperforms existing random-feature methods for spectrally localized kernels, enabling scalable, principled graph representation learning. Overall, this method offers a practical framework for accurate low-rank kernel approximations on large graphs, with particular advantage for narrow-band kernels that reflect global geometric structure.
Abstract
Node embeddings map graph vertices into low-dimensional Euclidean spaces while preserving structural information. They are central to tasks such as node classification, link prediction, and signal reconstruction. A key goal is to design node embeddings whose dot products capture meaningful notions of node similarity induced by the graph. Graph kernels offer a principled way to define such similarities, but their direct computation is often prohibitive for large networks. Inspired by random feature methods for kernel approximation in Euclidean spaces, we introduce randomized spectral node embeddings whose dot products estimate a low-rank approximation of any specific graph kernel. We provide theoretical and empirical results showing that our embeddings achieve more accurate kernel approximations than existing methods, particularly for spectrally localized kernels. These results demonstrate the effectiveness of randomized spectral constructions for scalable and principled graph representation learning.