Node-Oriented Spectral Filtering

• Node-oriented spectral filtering is a method where individual nodes or regions receive tailored filter responses to capture local graph heterogeneity.
• Techniques include polynomial recursions, node-variant FIR filters, and holomorphic operator calculus to ensure precise localization and adaptive processing.
• Applications span heterophilic node classification, cross-scale graph classification, and image processing, yielding measurable improvements in accuracy and efficiency.

Node-oriented spectral filtering refers to a broad class of spectral graph filtering methods where the filter response is adapted or specialized at the level of individual nodes, neighborhoods, or regions, rather than being globally shared across the network. This paradigm encompasses both theory and practice in graph signal processing (GSP), spectral graph neural networks (GNNs), distributed network operators, and local spectral statistics. Node-oriented approaches allow adaptive, localized, or position-sensitive spectral processing, often to address limitations of global (node-invariant) filters, especially in heterogeneous or irregular networks.

1. Fundamentals of Spectral Filtering on Graphs

Spectral filtering on graphs is grounded in the spectral theory of a graph shift operator $S$—usually the (normalized) Laplacian $L = D^{-1/2}AD^{-1/2}$ or adjacency matrix $A$—which admits an eigendecomposition $S = U\Lambda U^T$, with $U$ orthonormal and $$\Lambda = \mathrm{diag}(\lambda_1,\ldots,\lambda_N)$$. For a signal $x\in\mathbb R^N$ defined on $N$ nodes, the graph Fourier transform (GFT) is given by $\hat{x} = U^T x$, and a globally node-invariant spectral filter $h$ acts via $H = U h(\Lambda) U^T$, so that $Hx = U h(\Lambda) U^T x$.

Node-oriented spectral filtering breaks the global invariance: the filter $h$ is not universal, but is parameterized or adapted per node, per region, or localized subgraph. This generalization is essential for encoding heterogeneous patterns, local structures, or position-dependent processing (Zheng et al., 2022, Guo et al., 2023, Segarra et al., 2015).

2. Mathematical Framework and Node-specific Designs

The most general node-oriented spectral filter may assign to each node $i$ its own filter response $h_i(\lambda)$:

$$z_i = [U\,\mathrm{diag}(h_i(\lambda_1),\ldots,h_i(\lambda_N))\,U^T x]_i.$$

Alternatively, in the spatial domain for a polynomial basis $\{p_k(\lambda)\}$,

$z_i = \sum_{k=0}^K \eta_{i,k}\,[p_k(L) x]_i,$

where the coefficients $\eta_{i,k}$ are node-dependent (Zheng et al., 2022). This formulation admits further global-local trade-offs through low-rank reparameterizations (e.g., $\Psi = H\Gamma^T$ with $H_{i:}$ global features and $\Gamma$ global polynomial bases).

Distributed implementations in networked systems use node-variant FIR filters:

$$H_{nv} = \sum_{\ell=0}^{L-1} H^{(\ell)}S^\ell,\quad H^{(\ell)} = \mathrm{diag}(h^{(1)}_\ell,\ldots,h^{(N)}_\ell)$$

Each node $i$ can have its own filter coefficients $h^{(i)}_\ell$ (Segarra et al., 2015).

A variety of generalizations exist:

• Local spectral filtering within $K$-hop balls enables purely local (ego-network) filter estimation and transfer using rooted-ball distributions (Roddenberry et al., 2022).
• In heterogeneous GNNs, filters are constructed as convex combinations of global (shared) and local (node-/region-specific) responses; e.g., $\beta_{k,i} = \gamma_k\theta_{k,i}$ (Guo et al., 2023).
• Holomorphic functional calculus enables node-oriented spectral filters for directed graphs, using Jordan-Chevalley theory (Koke et al., 2023).

3. Localization, Generalized Translation, and Local Adaptation

Localization, a central property, ensures that a $K$-th order polynomial filter is $K$-hop local:

$[p_k(L)x]_i \text{ depends only on $x$in the$K$-neighborhood of$i$}.$

For node-adapted spectral kernels, generalized translation operators $T_i(g)$ “center” the spectral kernel around node $i$:

$$T_i(g) = \sqrt{n} \sum_{\ell=1}^n u_\ell u_\ell^T(i) g(\lambda_\ell)$$

ensuring node-localized responses (Zheng et al., 2022).

In transfer learning and distributional perspectives, local spectral densities (e.g., spectral moments of $K$-hop balls) serve as sufficient statistics for node-oriented filter design and matching across graphs (Roddenberry et al., 2022).

4. Practical Implementations and Algorithmic Aspects

Node-oriented spectral filters can be implemented using:

• Polynomial recursions (e.g., Chebyshev approximation, shifted Laplacians) for each node’s set of filter coefficients (Zheng et al., 2022, Guo et al., 2023, Cheung et al., 2018, Gadde et al., 2013).
• Distributed algorithms with per-node filter updates and aggregation; locality enables parallelism and scalability (Segarra et al., 2015).
• Spectrum-free computation of rational or polynomial filters using sparse systems, sidestepping explicit eigendecomposition (Patanè, 2020).

In GNNs, integration is typically via lightweight reparameterizations (e.g., mapping node embeddings to per-node filter coefficients with MLPs, or dynamically gating between local/global contributions) (Zheng et al., 2022, Guo et al., 2023, Guo et al., 17 Jan 2024).

For nonhomogeneous graphs, position-aware embeddings (e.g., learned node positions $P$) are used to parameterize node-specific filter weights (Guo et al., 2023).

5. Applications, Empirical Effects, and Theoretical Gains

Node-oriented spectral filtering has demonstrated superior or robust performance in:

• Heterophilic node classification: e.g., NFGNN and DSF variants show accuracy improvements of $1$–$5\%$ on strongly heterophilic datasets (Chameleon/Squirrel) over global spectral GNNS (Zheng et al., 2022, Guo et al., 2023).
• Cross-scale graph classification: spectral poolings in wavelet or adaptive bases show large improvements, e.g., $+15.55\%$ accuracy in specially constructed cross-scale benchmarks [(Zhang et al., 31 Aug 2024), data only from abstract].
• Collaborative filtering: node-signal polynomial spectral filters (e.g., PolyCF) outperform GNN embeddings by $1$–$3\%$ in Recall@20 and NDCG@20 (Qin et al., 23 Jan 2024).
• Image denoising, compression, and segmentation: node-oriented wavelets and bilateral-type filters yield edge adaptivity and improved signal-to-noise at lower computational cost (Cheung et al., 2018, Gadde et al., 2013).

Interpretability is enhanced, as learned node-level filters often reflect coherent local or positional structure, boundary sensitivity, and regional adaptation (Guo et al., 2023).

Theoretically, node-oriented filters guarantee polynomial localization, universal approximation of equivariant functions under unitary functional shift symmetries, and transferability via rooted-ball distributions (Zheng et al., 2022, Lin et al., 3 Jun 2024, Roddenberry et al., 2022).

6. Comparative Analysis, Limitations, and Open Perspectives

Relative to node-invariant (global) filters, node-oriented spectral filtering introduces $N$-fold more parameters, facilitating the exact or near-exact realization of a larger class of linear and nonlinear operators (Segarra et al., 2015). This comes with a trade-off between expressivity/local adaptation and parameter complexity; practical implementations use low-rank decompositions or local embeddings to balance these aspects.

Contrary to misconceptions, node-oriented filters, when properly parameterized, do not destroy spatial localization; polynomial filters remain strictly $K$-hop local, and rational or generalized spectral filters can still be implemented efficiently via sparse linear systems. Extensions to directed graphs and infinite domains are enabled via holomorphic operator calculus (Koke et al., 2023).

Open questions include the optimal parameterization of node-adaptive filters for dynamic/time-varying graphs, extensions to graph-level tasks, efficient low-rank approximations at scale, and hybrid integration of node-oriented spectral and attention-based models (Guo et al., 2023, Guo et al., 17 Jan 2024, Roddenberry et al., 2022).

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Summary Table: Core Model Types for Node-oriented Spectral Filtering

Method	Node Parameterization	Localization
NFGNN (Zheng et al., 2022)	Per-node polynomial coefficients	Exact $K$-hop
DSF (Guo et al., 2023)	Global + node-specific components	Polynomial, adaptive
Node-variant FIR (Segarra et al., 2015)	Per-node FIR (time-domain weights)	Degree = filter order
Local GSP (Roddenberry et al., 2022)	Function of $K$-hop rooted-ball	Locally exact
HoloNets (Koke et al., 2023)	Holomorphic operator-based	By choice of polynomial/resolvent
PolyCF (Qin et al., 23 Jan 2024)	Signal-level, polynomial basis	Multi-eigenspace

All models enable fine-grained, node-resolved adaptation of spectral responses, with theoretical guarantees (localization, equivariance, expressivity), efficient implementations, and empirically validated improvements on heterogeneous graphs.