Spectral Encoder in Frequency-Guided Graph Learning

• Spectral encoder is a specialized component that leverages spectral properties of the graph Laplacian, using low- and high-pass filters to capture both smooth and non-smooth node signal characteristics.
• It employs feature-driven graph masking to construct homophilic and heterophilic Laplacians, thereby adapting its spectral filters to encode diverse connectivity patterns.
• Empirical studies show that integrating spectral encoders with frequency-guided masking improves node classification accuracy across various benchmark datasets.

A spectral encoder is a specialized architectural component designed to process graph-structured data by encoding features with respect to the spectral properties of the graph Laplacian. In the context of frequency-guided graph structure learning (FgGSL), the spectral encoder plays a central role in extracting discriminative node representations, particularly for challenging heterophilic graphs—graphs where connected nodes are likely to have dissimilar labels and where feature similarity is a weak indicator of structural affiliation (Raghuvanshi et al., 29 Dec 2025). Through a bank of pre-designed low- and high-pass graph filters applied to learned, feature-driven graph structures, the spectral encoder in FgGSL jointly leverages both homophilic and heterophilic connectivity patterns to inform downstream tasks such as node classification.

1. Core Method: Spectral Encoder within FgGSL

Within the FgGSL framework, the spectral encoder operates by processing two complementary Laplacians $\{L_\text{Ho}, L_\text{Ht}\}$, which are derived from homophilic and heterophilic adjacency matrices inferred via learnable, feature-driven masks. For each Laplacian, the encoder applies filter banks at multiple spectral scales:

• The low-pass filter bank $\{h_L^{(j)}\}_{j=2}^J$ operates on $L_\text{Ho}$, accentuating low-frequency (smooth) components associated with homophily.
• The high-pass filter bank $\{h_H^{(j)}\}_{j=2}^J$ operates on $L_\text{Ht}$, emphasizing high-frequency (non-smooth) components linked to heterophily.

Each filter is a fixed polynomial approximated as

$h^{(j)}(L)X = \sum_{d=0}^{K_j} \alpha_d L^d X$

where the coefficients $\{\alpha_d\}$ correspond to the filter's spectral response. The outputs from all filter scales and branches are concatenated, forming the final encoded node representation:

$$H = [h_L^{(2)}(L_\text{Ho})X \mid\ldots\mid h_L^{(J)}(L_\text{Ho})X \mid h_H^{(2)}(L_\text{Ht})X \mid\ldots\mid h_H^{(J)}(L_\text{Ht})X] \in \mathbb{R}^{N\times 2(J-1)F}$$

This multiresolution embedding is then fed into a linear classifier for prediction.

2. Feature-Driven Graph Masking and Laplacian Definition

The spectral encoder in FgGSL relies not on the raw adjacency but on learned, symmetric mask functions

$$S_{\theta_k}(x_u, x_v) = \sigma\bigl(\Phi_{\theta_k}(x_u)^\top \Phi_{\theta_k}(x_v)\bigr)$$

where $\Phi_{\theta_k}$ is a small MLP and $\sigma$ is the sigmoid nonlinearity. These masking functions yield weighted adjacency matrices $\hat{A}_\text{Ho}$ and $\hat{A}_\text{Ht}$, which are then normalized to obtain $L_\text{Ho}$ and $L_\text{Ht}$. This dual-branch construction allows the spectral encoder to flexibly align its frequency response to the task-driven discovery of both assortative and disassortative (heterophilic) graph patterns.

3. Filter Bank Specification and Frequency Modeling

The filter banks instantiated in the spectral encoder are defined at multiple spectral scales $j=2,\ldots,J$:

• The low-pass polynomial filters target eigenvalues $\lambda\approx 0$:

$$h_L^{(j)}(\lambda) = (0.5\,\lambda)^{2^{j-1}} - (0.5\,\lambda)^{2^j}$$

• The high-pass polynomial filters highlight $\lambda\approx 2$:

$$h_H^{(j)}(\lambda) = (1-0.5\,\lambda)^{2^{j-1}} - (1-0.5\,\lambda)^{2^j}$$

This design ensures that the spectral encoder extracts both global and local patterns, mapping them into the representation space, and is especially suitable for settings where task labels do not align with clusters in the original topology.

4. Optimization, Stability, and Supervision

The spectral encoder's output is supervised via a total loss

$$\ell_{\rm total} = \ell_{\rm CE}(\hat{Y}_{\rm train}, Y_{\rm train}) + \alpha\,\ell_\text{Ho}(\hat{A}_\text{Ho}, \hat{Y}) + \beta\,\ell_\text{Ht}(\hat{A}_\text{Ht}, \hat{Y})$$

where $\ell_\text{Ho}$ and $\ell_\text{Ht}$ are label-based structural losses encouraging the learned masks to recover, respectively, homophilic and heterophilic edge patterns. The losses are defined by cosine similarity between model predictions, weighted by the mask elements, penalizing edges that contradict desired label alignments.

Stability guarantees are provided for both the structural loss and the robustness of the filter bank to perturbations in the Laplacian. In particular, the structural loss converges to its ideal target as prediction error decreases, and each polynomial filter exhibits bounded Lipschitz robustness under Laplacian perturbations, ensuring encoding stability even with imperfect graph inference.

5. Empirical Effects and Comparative Analysis

Empirical studies on six heterophilic benchmark datasets (Texas, Wisconsin, Cornell, Squirrel, Actor, Chameleon) demonstrate that combining spectral encoders with frequency-guided masking leads to substantial improvements in node classification accuracy, consistently outperforming standard GNNs (GraphSAGE, GAT), MLP baselines, and dedicated heterophilic designs (H2GCN, Geom-GCN) (Raghuvanshi et al., 29 Dec 2025). Detailed ablation studies indicate that both masking and dual-frequency (low- and high-pass) spectral encoding are necessary for optimal performance. The learned representations exhibit pronounced separation between intra-class and inter-class embedding similarities, a property not found in the raw input features.

Model	Texas	Wisconsin	Cornell	Squirrel	Actor	Chameleon
FgGSL	0.94 ± 0.08	0.96 ± 0.05	0.94 ± 0.08	0.58 ± 0.09	0.41 ± 0.02	0.79 ± 0.09
GraphSAGE	0.74 ± 0.08	0.74 ± 0.08	0.69 ± 0.05	0.37 ± 0.02	0.34 ± 0.01	0.50 ± 0.01
GAT	0.52 ± 0.06	0.49 ± 0.04	0.61 ± 0.05	0.40 ± 0.01	0.27 ± 0.01	0.60 ± 0.02

6. Computational Complexity and Implementation Characteristics

For $N$ nodes and $F$-dimensional features, the masking stage incurs $\mathcal{O}(N^2 D)$ complexity for inner products (assuming a fully connected candidate graph), where $D$ is the mask MLP hidden dimension. Filter application per bank and per scale is $\mathcal{O}(N^2 F)$, yielding a total per-epoch complexity of $\mathcal{O}(J N^2 F)$ for $J$ filter scales. If dense polynomial powers are naively computed, the cost scales as $\mathcal{O}(J N^3)$, but practical implementations mitigate this via sparsification or truncated approximations. Memory requirements include $N \times N$ mask matrices and $N \times 2(J-1)F$ filtered feature storage.

Key hyperparameters include the number of filter scales ($J$), mask network width ($D$), structural loss weights ($\alpha,\beta$), and the Adam optimizer learning rate ($10^{-3}$).

7. Significance and Interpretative Implications

The spectral encoder, as instantiated in FgGSL, provides a principled, interpretable mechanism for exploiting both low- and high-frequency graph information in a supervised, task-driven manner. Its ability to process learned, task-aligned Laplacians via fixed polynomial filter banks enables robustness to graph perturbations and adaptability across homophilic and heterophilic domains. A plausible implication is that such spectral encoders may set a paradigm for future graph learning research where both data-driven topology inference and frequency selective filtering are essential for optimal representation power and generalization, especially in non-homophilic scenarios (Raghuvanshi et al., 29 Dec 2025).