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Data-Driven Graph Filters via Adaptive Spectral Shaping

Dylan Sandfelder, Mihai Cucuringu, Xiaowen Dong

TL;DR

The paper addresses the challenge of designing graph filters that adapt to heterogeneous, multi-peak Laplacian spectra. It introduces Adaptive Spectral Shaping, which learns a baseline kernel g_theta(lambda) and modulates it with K Gaussian factors to form a flexible, multi-peak spectrum, with filtering implemented via Chebyshev polynomial expansions for scalability. A transferable variant, TASS, freezes the learned baseline across graphs and adapts only lightweight per-graph shaping parameters to enable few-shot transfer. Across synthetic benchmarks, the approach yields lower reconstruction error than fixed banks and provides interpretable, component-wise spectral decompositions, while enabling positive cross-graph transfer under matched compute. The method offers compact, interpretable spectral modules that can be plugged into graph signal processing pipelines and GNNs, supporting scalable, cross-graph generalization.

Abstract

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.

Data-Driven Graph Filters via Adaptive Spectral Shaping

TL;DR

The paper addresses the challenge of designing graph filters that adapt to heterogeneous, multi-peak Laplacian spectra. It introduces Adaptive Spectral Shaping, which learns a baseline kernel g_theta(lambda) and modulates it with K Gaussian factors to form a flexible, multi-peak spectrum, with filtering implemented via Chebyshev polynomial expansions for scalability. A transferable variant, TASS, freezes the learned baseline across graphs and adapts only lightweight per-graph shaping parameters to enable few-shot transfer. Across synthetic benchmarks, the approach yields lower reconstruction error than fixed banks and provides interpretable, component-wise spectral decompositions, while enabling positive cross-graph transfer under matched compute. The method offers compact, interpretable spectral modules that can be plugged into graph signal processing pipelines and GNNs, supporting scalable, cross-graph generalization.

Abstract

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.
Paper Structure (9 sections, 9 equations, 3 figures, 2 tables)

This paper contains 9 sections, 9 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Proposed Model
  3. Setup and notation.
  4. Experiments
  5. Experimental setup
  6. Single-graph filter reconstruction
  7. Transfer across graphs and signal regimes (TASS)
  8. Ablations and practical notes
  9. Conclusion & Future Work

Figures (3)

  • Figure 1: TASS pipeline. Learn a reusable baseline kernel $g_\theta$ on a source; on the target, freeze $g_\theta$ and adapt only $(\mu,\gamma,a)$ for $K$ components to localize spectral emphasis.
  • Figure 2: Learned filter adaptations with increasing number of components $K$: (a) single component, (b) two component, and (c) three component filters.
  • Figure 3: Learned filter decompositions compared against ground truth components.