Truncated Chebyshev Graph Encoding (TCGE)

• TCGE is a spectral method that uses truncated Chebyshev polynomial expansions of the graph Laplacian to encode graph signals with controlled locality.
• It enables multi-scale message passing up to K hops, balancing expressivity and efficiency while maintaining linear computational complexity.
• TCGE extends to advanced applications like privacy-preserving quantum ML and digital ink recognition, offering robust and scalable implementations.

Truncated Chebyshev Graph Encoding (TCGE) is a spectral method for encoding the structure and features of graph-structured data using truncated expansions in Chebyshev polynomials of the graph Laplacian. It achieves localized, multi-scale filtering over graphs, with applications in graph neural networks, privacy-preserving quantum machine learning, digital ink recognition, and more. TCGE is fundamentally grounded in efficient recursive computation of Chebyshev polynomials of a properly scaled and normalized Laplacian, enabling linear-time message passing up to arbitrary graph distance $K$ hops, while providing flexibility to control locality, expressivity, and numerical stability.

1. Mathematical Foundations and Core Construction

TCGE leverages Chebyshev polynomials of the first kind, $T_k(x)$, recursively defined by

$$T_0(x) = 1,\quad T_1(x) = x,\quad T_k(x) = 2xT_{k-1}(x) - T_{k-2}(x) \text{ for } k \ge 2.$$

Given a graph $G=(V,E)$ with adjacency $A$ and degree $D=\mathrm{diag}(A\mathbf{1})$, the (symmetric normalized) Laplacian is

$L = I - D^{-1/2} A D^{-1/2}$

with spectrum in $[0,2]$. For spectral graph convolution, all eigenvalues are linearly rescaled to $[-1,1]$: $\tilde L = \frac{2}{\lambda_{\max}} L - I$ where $\lambda_{\max}$ is the largest eigenvalue of $L$. This scaling ensures numerical stability and compatibility with Chebyshev recursion, with all spectral operations valid over $[-1,1]$.

A $K$-th order truncated Chebyshev spectral filter encodes graph signal $X \in \mathbb{R}^{n \times d}$ with

$Y = \sum_{k=0}^K T_k(\tilde L) X W_k$

where $W_k$ are learnable weight matrices or scalars per polynomial order. Each $T_k(\tilde L)X$ propagates information from up to $k$-hop neighbors, enabling one layer to capture multi-scale (local and global) structure while maintaining strict $K$-hop locality. This construction is efficient: the $T_k(\tilde L)X$ terms are computed via the three-term recurrence, never requiring explicit formation of high-degree polynomials or eigendecomposition, resulting in computational cost $O(K|E|d)$ per layer (Ashrafi et al., 27 Nov 2025, Semlani et al., 2023, He et al., 2022).

2. Role of Truncation Order, Locality, and Expressivity

The truncation order $K$ is the central hyperparameter in TCGE, governing the size of the filter polynomial basis. It determines:

• Locality: $T_k(\tilde L)X$ depends only on nodes up to $k$ hops away. $K=1$ recovers standard GCN behavior; higher $K$ enables larger receptive fields, crucial for global context.
• Expressivity: Higher $K$ permits fitting more complex spectral filter shapes, capturing both low-pass and high-pass behaviors.
• Computational and Statistical Tradeoffs: Larger $K$ increases cost linearly and may risk over-smoothing (collapse of node representations) or overfitting to noise. For many applications, small $K$ (e.g., $K=2,5$) suffices to achieve empirical gains (Ashrafi et al., 27 Nov 2025, Semlani et al., 2023, He et al., 2022, Jiang et al., 2021).

Empirically, careful $K$ selection improves performance by balancing local-global propagation with stability and efficiency, as shown in domains from fMRI population graphs to high-energy physics jet tagging and text-based graphs.

3. Efficient Algorithms and Practical Implementations

Actual implementation of TCGE is based on recursive computation: $$\begin{align*} X_0 &= X \ X_1 &= \tilde L X \ X_k &= 2\tilde L X_{k-1} - X_{k-2}, \quad k\ge 2 \ Y &= \sum_{k=0}^K X_k W_k \end{align*}$$ where only two buffer matrices $X_{k-1}, X_{k-2}$ are needed in memory at each step. The matrix-matrix multiplication with sparse $\tilde L$ yields linear complexity in the number of edges and features. Weight sharing or independent weights per channel/filter order are available depending on architecture design (Ashrafi et al., 27 Nov 2025, Semlani et al., 2023, He et al., 2022).

Extensions include: concatenating multi-order outputs before nonlinearity, integrating batch normalization, and fusing TCGE within larger architectures (e.g., with GAT layers, MLPs, or dynamic cross-attention modules).

4. Extensions Beyond Classical Graph Neural Networks

Beyond standard graph convolutional architectures, TCGE generalizes to several advanced contexts:

• Chebyshev-Sobolev Graph Encoding: Extends the polynomial basis to Sobolev orthogonal polynomials, incorporating smoothness via weighted edge-difference (gradient) terms in the inner product. This provides compact, shape-aware graph embeddings, suitable for applications such as digital-ink recognition and signature verification. The expansion is:

$$\widehat f = \sum_{k=0}^N \theta_k S_{\lambda, k}(\tilde L) f$$

with $S_{\lambda, k}$ obtained via Gram–Schmidt on monomials w.r.t. a Sobolev inner product that combines node and edge terms (Kalhan et al., 4 Aug 2024).

• Quantum Information Encoding: In quantum ML, TCGE expresses classical data into quantum states via Chebyshev-parameterized $RY$ rotations entangled by a graph-state circuit (CZ ladder). The resulting n-qubit state encodes nonlinear Chebyshev features up to order $K$ distributed globally, providing privacy guarantees via entanglement and highly non-separable representations, and resisting snapshot inversion attacks in quantum machine learning (Zhang et al., 30 Nov 2025).

5. Comparison to Other Spectral and Message-Passing Methods

TCGE is typically contrasted with:

Method	Spectral Basis	Locality	Computational Cost	Key Limitations
Full Spectral	Arbitrary (eigen)	Global	$O(N^3)$ (eigendecomp)	No locality, infeasible for large $N$
Standard GCN	$K=1$ (linear)	1-hop	$O(|E|)$ per layer	Poor expressivity (only low-pass)
TCGE/ChebNet	Chebyshev, $K>1$	Up to $K$-hops	$O(K|E|)$ per layer	Risk of overfitting at high $K$
ChebNetII	Chebyshev interp.	Up to $K$-hops	$O(K|E|) + O(K^2)$	Improved, avoids coefficient pathology
HDGCN	Chebyshev + dyn.	Multi-hop	$O(K|E|d)$ (& attn)	Requires attention mechanism

TCGE retains strict $K$-hop locality, arbitrary filter expressivity up to degree $K$, and avoids expensive spectral decompositions, making it applicable to large-scale graphs and scalable architectures (Ashrafi et al., 27 Nov 2025, He et al., 2022, Jiang et al., 2021).

ChebNetII resolves the "illegal coefficients" pathology and Runge phenomenon by using Chebyshev interpolation at carefully selected nodes, enforcing monotonic decay of polynomial coefficients and yielding better minimax approximation properties (He et al., 2022). Dynamic variants, such as HDGCN, replace fixed high-order convolutions by data-driven multi-vote attention modules, mitigating over-smoothing while enabling efficient multi-hop aggregation (Jiang et al., 2021).

6. Empirical Performance and Application Domains

TCGE has demonstrated empirical success in various domains:

• Neuroimaging-based Disorder Classification: Integration of multimodal MRI data using multi-branch TCGE-based GCNs achieves enhanced accuracy and AUC compared to conventional baselines (Ashrafi et al., 27 Nov 2025).
• Jet Tagging in High-Energy Physics: Truncated Chebyshev filters provide significant accuracy improvements over classical GNNs and other taggers by efficiently encoding multi-particle interactions in jets (Semlani et al., 2023).
• Textual and Large-Scale Inductive Tasks: High-order dynamic Chebyshev methods in HDGCN outperform standard GCN, GAT, and Transformer-based models on NLP and node classification benchmarks, especially in data-limited regimes (Jiang et al., 2021).
• Quantum Machine Learning: In DyLoC, TCGE applied at the input layer creates a robust privacy barrier, dramatically increasing adversary inversion error under snapshot attacks, with only $O(1)$ circuit depth and resource overhead (Zhang et al., 30 Nov 2025).
• Digital Ink Analysis: Chebyshev-Sobolev encodings result in more compact and class-separable coefficient representations for online handwriting, improving $k$-NN and clustering performance while retaining interpretability (Kalhan et al., 4 Aug 2024).

7. Theoretical Guarantees, Limitations, and Future Directions

TCGE inherits favorable theoretical properties from its Chebyshev polynomial foundation: minimax approximation rates for smooth filters, numerical stability under spectral scaling, and efficient linear-time graph propagation. ChebNetII provides explicit coefficient decay and reduces oscillatory artifacts (Runge phenomenon) that can cause overfitting in unconstrained Chebyshev expansions (He et al., 2022).

Over-smoothing remains a challenge at high truncation order; recent solutions involve dynamic or adaptive filtering (MVCAttn, Chebyshev interpolation). In privacy-constrained quantum ML, TCGE-induced ruggedness in the loss landscape provably blocks analytic inversion for practical adversaries (Zhang et al., 30 Nov 2025). For graph-structured data beyond Euclidean domains, Chebyshev-Sobolev extensions provide an interpretable and efficient basis for both functional and structural aspects (Kalhan et al., 4 Aug 2024).

A plausible implication is increased adoption of TCGE and its variants in settings requiring balance between expressive signal propagation, computational tractability, controllable locality, and (in quantum/classical) privacy or robustness constraints. Open directions include integration with attention mechanisms, non-Euclidean manifold graphs, and automated selection of truncation order and basis adaptations for heterogeneous graphs.

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References:

(Ashrafi et al., 27 Nov 2025, He et al., 2022, Zhang et al., 30 Nov 2025, Semlani et al., 2023, Jiang et al., 2021, Kalhan et al., 4 Aug 2024)