# Asymmetry-Aware Bochner–Riesz Graph Filter

- ID: 2039
- Canonical URL: https://synthcore.org/idea/2039/asymmetry-aware-bochner-riesz-graph-filter
- API JSON: https://synthcore.org/api/idea/2039.json
- API Markdown: https://synthcore.org/api/idea/2039.md
- Verification status: unverified
- Source: [arXiv:2608.09247](https://arxiv.org/abs/2608.09247)
- Category: architecture
- Solves: stability, accuracy, generalization
- ML areas: graph-nn, regularization
- Math tags: harmonic-analysis, spectral-theory, geometry, approximation-theory
- Ratings: usefulness 5/10; difficulty 5/10; novelty 6/10

## Idea description

Replace a sharp graph-Laplacian spectral filter with a Bochner–Riesz filter whose smoothness exponent increases when the graph contains regions with different effective dimensions. Estimate the largest local dimension and dimension gap from neighborhood growth, then choose the exponent above both the classical spectral threshold and the asymmetric obstruction threshold. This should suppress unstable high-frequency mixing in heterogeneous graphs while preserving more low-frequency signal than an aggressively Gaussian filter.

## Mathematical statement

Lemma 6.3 gives the classical condition $\delta>D|1/p-1/2|-1/2$, with the lower bound clipped at zero, and an asymmetric obstruction $\delta<|d_1-d_2||1/p-1/2|$. Here $p\in[1,\infty]$ is the operator norm exponent, $d_1,d_2>1$ are effective dimensions of the two ends, $D=d_1\vee d_2$ is their maximum, and $\delta\ge0$ is the Bochner–Riesz smoothness order. A conservative adapted threshold is $\delta>\max(D|1/p-1/2|-1/2,0,|d_1-d_2||1/p-1/2|)+\varepsilon$, with margin $\varepsilon>0$. The spectral multiplier is $m_{\Lambda,\delta}(\lambda)=(1-\lambda/\Lambda)_+^\delta$, where $\lambda$ is a graph-Laplacian eigenvalue, $\Lambda$ is a cutoff, and $(t)_+=\max(t,0)$. For heterogeneous graphs, replace $D$ by a robust maximum local dimension and $|d_1-d_2|$ by a robust local dimension gap. The paper proves this phenomenon for a one-dimensional manifold with unequal ends; applying it to graphs is a testable transfer hypothesis rather than a direct theorem.

## Key formulas

- $$\delta>\max\left(D\left|\frac{1}{p}-\frac{1}{2}\right|-\frac{1}{2},0\right),\qquad D=d_1\vee d_2.$$
- $$\delta<|d_1-d_2|\left|\frac{1}{p}-\frac{1}{2}\right|\quad\Longrightarrow\quad \sigma_1^\delta(\Delta_{d_1,d_2})\text{ is unbounded in }L^p(\widetilde{\mathbb R}).$$
- $$m_{\Lambda,\delta}(\lambda)=\left(1-\frac{\lambda}{\Lambda}\right)_+^\delta,\qquad \delta_{\mathrm{safe}}=\max\left(D\left|\frac{1}{p}-\frac{1}{2}\right|-\frac{1}{2},0, G\left|\frac{1}{p}-\frac{1}{2}\right|\right)+\varepsilon.$$
- $$X'=m_{\Lambda,\delta}(L)X=\left(I-\frac{L}{\Lambda}\right)_+^\delta X.$$

## Implementation notes

Integrate this at a graph-convolution or message-passing layer, replacing the usual polynomial spectral filter or near-hard low-pass filter. Let $L$ be the normalized graph Laplacian and $X\in\mathbb R^{n\times c}$ the node-feature matrix. First estimate local effective dimension: for each node $v$, compute neighborhood counts $N_v(r)$ at several radii and fit the slope $d_v$ of $\log N_v(r)$ against $\log r$ by least squares. Set $D$ to a robust upper percentile of the $d_v$ values and set $G$ to the difference between robust upper and lower percentiles. Choose $p=1$ or $p=\infty$ for a conservative filter, giving a threshold approximately $\max(D/2-1/2,G/2)+\varepsilon$; also test $p=2$, for which the dimension-dependent threshold vanishes. Approximate $m_{\Lambda,\delta}(L)X$ without eigendecomposition using $K$ Chebyshev terms. Rescale $L$ to $\widetilde L\in[-1,1]$, fit coefficients for $(1-\lambda/\Lambda)_+^\delta$, and evaluate $T_0(X)=X$, $T_1(X)=\widetilde L X$, and $T_{k+1}(X)=2\widetilde L T_k(X)-T_{k-1}(X)$. The paper supplies the threshold and multiplier; local dimension estimation, cutoff $\Lambda$, polynomial degree $K$, and margin $\varepsilon$ are empirical. Begin with a 2-layer GCN on graphs containing dense and sparse communities, comparing a fixed hard cutoff, fixed Gaussian filter, and adaptive Bochner–Riesz filtering at equal FLOPs. Track validation accuracy, feature oversmoothing, gradient-norm variance, and robustness to random edge deletion. Success means higher accuracy with lower feature collapse or gradient variance and no material inference-cost increase over the Chebyshev baseline.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.