JacobiConv: Spectral GNN with Jacobi Polynomials

• JacobiConv is a linear spectral GNN model that employs tunable Jacobi polynomials for adaptive filter design, aligning with the graph signal distribution.
• The model integrates a small feature mixing layer before spectral filtering, leading to a convex loss surface and efficient optimization.
• Empirical evaluations demonstrate that JacobiConv outperforms traditional nonlinear models on node classification benchmarks, achieving up to 12% higher accuracy.

JacobiConv is a linear spectral Graph Neural Network (GNN) model that leverages the orthogonality and adaptability of Jacobi polynomials as the basis of its spectral filters. By selectively matching the spectral filter basis to the empirical density of graph signals using tunable Jacobi polynomials, JacobiConv achieves both theoretical universality and state-of-the-art empirical performance on diverse node classification and filtering benchmarks—despite eschewing pointwise nonlinearities and multilayer perceptrons entirely (Wang et al., 2022).

1. Foundational Concepts in Spectral Graph Neural Networks

Spectral GNNs perform convolution based on the eigendecomposition of the graph Laplacian, viewing node features $X \in \mathbb{R}^{n \times d}$ as signals defined over graph vertices. Traditional polynomial spectral filters act as $Y = g(L)X = U g(\Lambda) U^\top X$, where $L=U \Lambda U^\top$ is the normalized Laplacian. In Chebyshev-based models, the filter is parameterized as a truncated expansion $Y = \sum_{k=0}^K \theta_k T_k(\tilde{\Lambda}) X$, with $T_k$ denoting Chebyshev polynomials evaluated on rescaled spectrum $\tilde{\Lambda} = 2\Lambda - I$. The operation is equivalently applied in the vertex domain as $Y = \sum_{k=0}^K \theta_k L^k X$. This formulation enables learning arbitrary polynomial spectral filters but typically relies on fixed polynomial families and may introduce nonlinearity or post-filter feature mixing.

2. JacobiConv Model Architecture and Polynomial Specialization

JacobiConv maintains the general polynomial spectral filtering blueprint—eschewing nonlinearity—and introduces the Jacobi polynomial basis $\{P_k^{(\alpha, \beta)}(\hat{A})\}$, where $\hat{A} = I - L$ is the normalized adjacency. For an output $Z \in \mathbb{R}^{n \times d'}$, each dimension $\ell=1,\ldots,d'$ comprises a distinct filter:

$$Z_{:, \ell} = g_\ell(\hat{A})X = \sum_{k=0}^K \theta_{k, \ell} P_k^{(\alpha, \beta)}(\hat{A}) X.$$

Feature mixing via a small fully connected layer and bias (i.e., $X' = XW + b$) precedes spectral filtering. The Jacobi polynomial basis, parameterized by exponents $(\alpha, \beta)$, is recursively constructed and can be adapted to match the signal distribution in the graph spectrum, optimizing the expressive utility of the filter basis.

3. Properties and Implementation of Jacobi Polynomials

Jacobi polynomials $P_k^{(\alpha, \beta)}(x)$ are defined on $[-1, 1]$ by the recurrence:

• $P_0^{(\alpha, \beta)}(x) = 1$,
• $$P_1^{(\alpha, \beta)}(x) = \frac{1}{2}[(2+\alpha+\beta)x + (\alpha-\beta)]$$,
• higher degrees are recursively generated and maintain the orthogonality relation

$$\int_{-1}^1 (1-x)^\alpha (1+x)^\beta P_m^{(\alpha, \beta)}(x) P_n^{(\alpha, \beta)}(x) dx = 0 \quad \text{for } m \neq n.$$

In graph applications, these polynomials are evaluated on the normalized adjacency $\hat{A}$, inheriting orthogonality properties that depend on the empirical graph Fourier spectral density $$f(\lambda) = \sum_i \| (U^\top X)_{i, \cdot} \|^2 \delta(\lambda - \lambda_i)$$. Adaptive tuning of $(\alpha,\beta)$ enables JacobiConv to closely align filter bases with graph signal characteristics.

4. Expressive Power and Universality

A central theorem shows that, on a fixed graph without repeated Laplacian eigenvalues and with node features whose graph-Fourier transform has full spectral support, a single linear JacobiConv layer of sufficiently high degree ($K \geq n-1$) can realize any one-dimensional node-wise mapping $Z \in \mathbb{R}^n$. These two conditions—uniqueness in Laplacian spectrum and spectral completeness of node features—guarantee universality in the spectral sense. Notably, these same conditions ensure that the 1-Weisfeiler–Lehman (1-WL) test is maximally expressive on the graph, establishing a direct correspondence between spectral and spatial GNN expressive power (Wang et al., 2022).

5. Filter Learning, Convexity, and Optimization

JacobiConv’s filter parameters $\theta_{k,\ell}$ appear linearly in the model. Consequently, the loss surface is convex with respect to these parameters. The Hessian (with respect to the filter vector) converges to a diagonal dominated structure if the Jacobi basis is tuned to the empirical spectral density, theoretically enabling optimal quadratic convergence under standard gradient methods. Filter construction leverages a three-term recurrence, avoiding explicit Gram–Schmidt or cumbersome matrix operations. An optional polynomial coefficient reparameterization can further enhance conditioning.

6. Empirical Evaluation and Benchmarks

Experiments demonstrate JacobiConv’s superior empirical performance:

• Synthetic spectral regression. Using ground-truth spectral targets (e.g., low-pass, high-pass, band-pass, reject, comb filters) on 2D image grids, JacobiConv consistently attains the lowest MSE—often by an order of magnitude—across all filter types compared to baselines (ChebyNet, GPRGNN, ARMA, BernNet, and monomial/Chebyshev/Bernstein/Jacobi linear PFME variants).
• Node classification. On ten datasets—Cora, CiteSeer, PubMed, Amazon Computers, Amazon Photo, Chameleon, Actor, Squirrel, Texas, and Cornell—JacobiConv, as a single linear layer of degree $K \approx 10$, outperforms nonlinear models (GCN, APPNP, ChebyNet, GPRGNN, BernNet) on nine of ten, achieving up to 12% higher accuracy (notably on Squirrel). Ablation reveals that using multiple filter outputs per channel, Jacobi basis selection, and coefficient decomposition each confer empirical benefits of 0.5–1.5% accuracy. The introduction of an extra two-layer ReLU MLP does not surpass the performance of a residual linear variant, reinforcing that explicit nonlinearity is unnecessary given a well-adapted spectral basis.

7. Computational Complexity and Scalability

JacobiConv incurs computational cost $O(Kmd)$ for an order-$K$ spectral layer over a graph with $m = |E|$ edges and feature dimension $d$, matching ChebyNet and GPRGNN. BernNet’s use of the Bernstein basis induces $O(K^2md)$ complexity. JacobiConv’s overhead is limited to the three-term recurrence and storage for $K+1$ scalars per output channel, two possible basis exponents $(\alpha, \beta)$, and the parameter matrix $W$ for feature mixing. In regimes where $d \gg K$, the parameter count is one-tenth that of APPNP or BernNet. Per-epoch runtime remains within 10–20% of the fastest baselines reported (Wang et al., 2022).

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JacobiConv exemplifies a rigorous linear spectral GNN, leveraging the Jacobi polynomial basis to match the empirical spectral distribution of graph signals, thereby achieving provable universality and strong empirical accuracy in classification and filtering tasks on diverse synthetic and real-world graphs, all without recourse to nonlinearities or MLPs (Wang et al., 2022).