Data-Driven Graph Filters via Adaptive Spectral Shaping

We introduce Adaptive Spectral Shaping, a data-driven framework for graph filtering that learns a reusable baseline spectral kernel and modulates it with a small set of Gaussian factors. The resulting multi-peak, multi-scale responses allocate energy to heterogeneous regions of the Laplacian spectrum while remaining interpretable via explicit centers and bandwidths. To scale, we implement filters with Chebyshev polynomial expansions, avoiding eigendecompositions. We further propose Transferable Adaptive Spectral Shaping (TASS): the baseline kernel is learned on source graphs and, on a target graph, kept fixed while only the shaping parameters are adapted, enabling few-shot transfer under matched compute. Across controlled synthetic benchmarks spanning graph families and signal regimes, Adaptive Spectral Shaping reduces reconstruction error relative to fixed-prototype wavelets and learned linear banks, and TASS yields consistent positive transfer. The framework provides compact spectral modules that plug into graph signal processing pipelines and graph neural networks, combining scalability, interpretability, and cross-graph generalization.