Random Wavelet Features for Graph Kernel Machines

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.

💡 Research Summary

The paper tackles the long‑standing scalability bottleneck of graph kernels, which provide principled similarity measures between nodes or sub‑graphs but require O(n²) time and memory to compute directly on large networks. Inspired by random feature (RF) methods that approximate shift‑invariant kernels in Euclidean spaces, the authors propose Random Wavelet Features (RWF), a randomized spectral embedding that approximates any given graph kernel in a low‑dimensional Euclidean space.

The construction starts from the eigendecomposition of the graph Laplacian L = UΛUᵀ. For a target kernel defined by a spectral filter g(·) (e.g., Heat, Diffusion, p‑step random walk), the filter is applied to the eigenvalues, yielding g(Λ)Uᵀδ_i for each node i (δ_i is the i‑th canonical basis vector). D random Gaussian vectors ξ₁,…,ξ_D are sampled, and each node’s embedding is formed as φ_i = √(2/D)