Asymmetry-Aware Bochner–Riesz Graph Filter

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.