Reconsidering the Spectral Story in Graph Neural Networks

For nearly a decade, the field of graph representation learning has carried two parallel narratives about how its models work. One is the message-passing story, where nodes iteratively aggregate information from their neighbors. The other is the spectral story, where graph signals are decomposed into frequency components and filtered in a transformed domain. The second narrative has always borrowed prestige from classical signal processing, lending an air of mathematical depth to graph convolution. A new position paper, "Position: Spectral GNNs Are Neither Spectral Nor Superior for Node Classification" by Jiang, Wang, Lones, Chen, and Pang, argues that this borrowed prestige is largely unearned. The authors contend that for node classification, spectral graph neural networks neither meaningfully capture a graph spectrum nor reliably outperform simpler message-passing alternatives.

I want to walk through the core arguments, separate the parts I find genuinely convincing from the parts that need qualification, and then offer some thoughts on where this leaves the field.

Two Foundational Glitches

The paper identifies two theoretical problems that, taken together, undermine the standard justification for spectral GNNs.

The first concerns the so-called graph Fourier basis. The standard construction defines a Graph Fourier Transform through the eigen-decomposition of the normalized Laplacian, $\hat{L} = U \Lambda U^\top$, and then treats the eigenvectors in $U$ as the graph's Fourier modes. The paper's objection is that this construction produces a valid orthonormal basis for the operator $\hat{L}$, but it does not inherit the semantics that make Fourier analysis meaningful in Euclidean settings.

This is the crucial point, and it is worth stating carefully. In classical signal processing, the Fourier modes are not just any eigenbasis. They are the characters of a translation group. On $\mathbb{Z}_n$, the discrete Fourier transform diagonalizes the cyclic shift operator, and its modes are globally meaningful oscillations on a regular grid. The notion of frequency is tied to translation invariance. General graphs, however, have no canonical translation operator and no globally consistent set of orthogonal directions. The paper's analogy is apt: using Laplacian eigenvectors as a graph's Fourier basis is like abandoning the standard 2D Fourier transform of an image and instead diagonalizing a pixel-connectivity Laplacian, then calling its eigenvectors "spectral components." You get an orthonormal basis, but you lose the property that made the original transform interpretable.

I find this argument persuasive, and importantly, the authors sharpen a prior position by Guo et al. (2025). Guo et al. attributed the problem to a continuous-versus-discrete mismatch. Jiang and colleagues correctly reject that framing, since Fourier analysis extends cleanly to discrete domains via the DFT. The real issue is structural: the absence of a global shift operator on irregular graphs. This is a more precise diagnosis, and it relocates the problem from a non-issue (discretization) to the actual culprit (lack of translation symmetry).

The second glitch targets the polynomial approximation narrative. Spectral GNNs avoid the cost of eigen-decomposition by approximating spectral filters with polynomials of the Laplacian, using Chebyshev, Bernstein, or similar bases. The selling point is that higher-order polynomials approximate richer filters. The paper points out a subtle but damaging fact: on an $n$-node graph, an $(n-1)$-th order polynomial can exactly interpolate any ideal spectral response through a Vandermonde system. This is not approximation at all; it is exact determination once the polynomial order reaches $n-1$.

The implication is twofold. First, the framing of "approximation quality improving with order $K$" is theoretically confused, since the limit is exact interpolation rather than convergence to some target. Second, and more interesting, a $k$-th order polynomial in $\hat{A}$ exactly recovers $k$-hop message passing. So the practical effect of a spectral filter is identical to running message passing over a $k$-hop neighborhood. The spectral wrapper adds nothing operationally that message passing does not already provide.

Where the Low-Pass Behavior Actually Comes From

The most widely repeated claim in this literature is that the graph convolutional network (GCN) works because it acts as a low-pass filter, smoothing node features across the graph. The paper argues that this low-pass and high-pass behavior is real, but its explanation is wrong. The filtering behavior arises directly from message-passing dynamics, that is, from repeated neighbor averaging and its tendency toward oversmoothing, rather than from any Graph Fourier Transform argument. The authors claim that a GFT-based argument alone cannot establish the filtering interpretation.

This reframing matters because it separates an empirical observation (features get smoothed) from a theoretical justification (the model is performing frequency-domain filtering). The observation is correct; the justification is decorative. If message passing already explains why features smooth out, then invoking the spectrum is an unnecessary layer of interpretation.

The Directed Case: MagNet and HoloNet

The paper's most concrete and falsifiable claim concerns two directed spectral models, MagNet (Zhang et al., 2021) and HoloNet (Koke and Cremers, 2024). These models use magnetic Laplacians and complex-valued constructions to handle edge directionality, and they report strong results. The authors argue that this strength is not spectral in origin. Instead, they attribute it to implementation choices that effectively reduce the models to powerful message-passing networks. When the models are re-implemented to be consistent with their stated spectral algorithms, the authors report that performance drops substantially.

This is a serious accusation, and it is the kind of claim that the community can and should verify directly. If correct, it means the reported gains came from accidental departures from the spectral design rather than from the spectral machinery itself. It would be a textbook example of a benchmark result being explained by an implementation detail rather than the intended mechanism.

My Take

I think the structural argument about translation symmetry is the strongest contribution here, and it deserves to outlast the rest of the paper. The graph community has been loose with terminology for years, and calling the Laplacian eigenbasis a "Fourier basis" has done real conceptual damage by implying properties that simply do not hold. Naming things accurately is not pedantry; it shapes which research directions seem promising.

That said, I would push back on the framing in one respect. The Vandermonde interpolation observation, while technically correct, applies at order $n-1$, which is far beyond the orders used in practice (typically $K$ around 2 to 10). In the practical regime, polynomial filters genuinely are low-dimensional approximations. So the "no approximation is happening" claim is sharpest as a statement about the asymptotic limit, less so about deployed models. The deeper point survives, though: a low-order polynomial filter and $k$-hop message passing are the same object, so the spectral language is redundant either way.

There is also a tension I want to flag. If spectral GNNs are equivalent to MPNNs, then the failure to outperform is unsurprising, but neither is the failure damning. Equivalence cuts both ways. A spectral parameterization might still offer a useful inductive bias or a convenient way to design filters, even if it is not doing anything a sufficiently expressive MPNN could not do. The paper's stronger claim is that the spectral framing actively misleads, and I think that is the right hill to defend.

Open Questions

The most immediate empirical question is whether the MagNet and HoloNet re-implementation results replicate independently. If they do, it would be a strong case study in how benchmark leaderboards reward implementation artifacts.

A second question is whether anything is lost by abandoning spectral language entirely. Spectral analysis still offers tools for reasoning about oversmoothing, expressivity, and signal propagation, even if "frequency" on a graph lacks its classical meaning. The challenge is to keep the analytical tools while dropping the misleading semantics.

My own prediction echoes the one in the accompanying summary: over the next few years, the spectral branch of graph learning will be progressively re-described in message-passing terms, and the directed spectral models will be re-derived as plain MPNNs once their implementations are made consistent. The spectrum was always a useful metaphor. It was just never the mechanism.