# Two-Scale Parabolic Filter Block

- ID: 424
- Canonical URL: https://synthcore.org/idea/424/two-scale-parabolic-filter-block
- API JSON: https://synthcore.org/api/idea/424.json
- API Markdown: https://synthcore.org/api/idea/424.md
- Verification status: failed_benchmark
- Source: [arXiv:2607.05181](https://arxiv.org/abs/2607.05181)
- Category: architecture
- Solves: stability, scalability, memory
- ML areas: ssm, attention, graph-nn, memory
- Math tags: pde, harmonic-analysis, functional-analysis, geometry
- Ratings: usefulness 7/10; difficulty 6/10; novelty 6/10

## Idea description

Construct a spatiotemporal neural block from localized functions of a learned parabolic operator instead of unrestricted attention or convolution. Use one filter for fine-scale diffusion and another for coarse-scale temporal aggregation, with the scale ratio controlling information propagation. The block should suppress distant interactions while still permitting long-range mixing through coarse filters.

## Mathematical statement

The paper studies the heat-type operator \(\mathcal{H}u=\partial_tu-\operatorname{Div}_x(A(x,t)\nabla_xu)\), where \(A(x,t)\) is a bounded, possibly rough coefficient field. It defines localized holomorphic filters \(\psi(z)=z^{3\alpha}(1+z)^{-6\alpha}\) and \(\varphi(z)=(1-(1+z)^{-\beta})^{3\alpha}\), with \(\alpha>0\) and \(\beta\geq0\). Lemma 7.1 gives \(\|\mathbf{1}_F\psi(\lambda^2\mathcal{H})\varphi(r^2\mathcal{H})\mathbf{1}_Eu\|_p\leq C(1+d(E,F)/\min(\lambda,r))^{-6\alpha}\|\mathbf{1}_Eu\|_p\), where \(E,F\) are measurable input and output regions, \(d(E,F)\) is their parabolic distance, \(\lambda,r>0\) are filter scales, \(p\in[1,2]\), and \(C\) is independent of \(E,F,\lambda,r\). The lemma also gives \(\|\psi(\lambda^2\mathcal{H})\varphi(r^2\mathcal{H})u\|_p\leq C\min(1,(r/\lambda)^{2\alpha})\|u\|_p\). The adaptation uses the explicit off-diagonal decay to limit distant interactions and the scale-ratio factor to attenuate incompatible resolutions.

## Key formulas

- $$\mathcal{H}u=\partial_tu-\operatorname{Div}_x(A(x,t)\nabla_xu)$$
- $$\psi(z)=z^{3\alpha}(1+z)^{-6\alpha},\qquad \varphi(z)=\bigl(1-(1+z)^{-\beta}\bigr)^{3\alpha}$$
- $$\left\|\mathbf{1}_{F}\psi(\lambda^{2}\mathcal{H})\varphi(r^{2}\mathcal{H})\mathbf{1}_{E}u\right\|_{p}\leq C\left(1+\frac{d(E,F)}{\min(\lambda,r)}\right)^{-6\alpha}\left\|\mathbf{1}_{E}u\right\|_{p}$$
- $$\left\|\psi(\lambda^{2}\mathcal{H})\varphi(r^{2}\mathcal{H})u\right\|_{p}\leq C\min\left(1,\left(\frac{r}{\lambda}\right)^{2\alpha}\right)\|u\|_{p}$$

## Implementation notes

(1) Integration point: replace one dense spatiotemporal-attention block in a video transformer, video SSM, or spatiotemporal graph neural network. Represent features as \(X\in\mathbb{R}^{B\times T\times N\times d}\), where \(T\) is the number of frames and \(N\) is the number of spatial patches or graph nodes. Build a sparse spatial diffusion matrix \(L_A\) from local edges with nonnegative conductances \(A_{ij}\), and a backward temporal difference operator \(D_t\). Define \(H=D_t+L_A+\epsilon I\), with \(\epsilon>0\) for numerical invertibility. (2) Pseudocode: choose scales \(\lambda_1<\cdots<\lambda_K\), set \(r_k=\lambda_{k-1}\), and for every scale compute \(Y_k=\psi(\lambda_k^2H)\varphi(r_k^2H)X\). Approximate each inverse power \((I+sH)^{-q}\) by a short Chebyshev polynomial or 5--20 conjugate-gradient iterations; when \(3\alpha\) is integral, apply \(H\) repeatedly for the numerator. Concatenate or sum \(Y_k\), apply a learned channel projection, and use \(X_{out}=X+W_o[Y_1\|\cdots\|Y_K]\). Normalize scale gates with a softmax. (3) Use the paper's filter shapes, the scale attenuation \(\min(1,(r/\lambda)^{2\alpha})\), and the off-diagonal exponent \(6\alpha\). Estimate filter norms by power iteration and use \(d((x,t),(y,s))=\max(\|x-y\|,\sqrt{|t-s|})\) for irregular video sampling. (4) First experiment: train a small Moving-MNIST predictor or UCF101 classifier with 16 frames and 64 spatial patches. Compare against local attention, full attention, and temporal convolution at equal width and measured FLOPs. Success means lower validation loss at equal FLOPs, stable gradients under noisy \(A\), and lower activation memory than full attention. Mask one region and test whether influence decays approximately as \((1+d/\min(\lambda,r))^{-6\alpha}\).

## Verification

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: inconclusive
- Verdict: Built a differentiable two-scale parabolic spectral filter block in parabolic_block.py and numerical verification scripts. The mechanism manifested: the exact small-r response exponent was predicted as 3.0 and observed as 2.996, while the off-diagonal decay exponent was predicted as -3.0 and observed as -3.095. The theorem-style scale bound also held with maximum observed gain/bound ratio 0.0567; a tiny regression gave lower MSE for the idea (0.001162 vs 0.001317) but at substantially higher dense-operator cost, so this is evidence for the mechanism rather than a demonstrated practical win.

### Mechanism check

- Verdict: Built a differentiable two-scale parabolic spectral filter block in parabolic_block.py and numerical verification scripts. The mechanism manifested: the exact small-r response exponent was predicted as 3.0 and observed as 2.996, while the off-diagonal decay exponent was predicted as -3.0 and observed as -3.095. The theorem-style scale bound also held with maximum observed gain/bound ratio 0.0567; a tiny regression gave lower MSE for the idea (0.001162 vs 0.001317) but at substantially higher dense-operator cost, so this is evidence for the mechanism rather than a demonstrated practical win.
- Confidence: 8/10
- Limitations: The implementation uses dense eigendecomposition rather than the planned Chebyshev or conjugate-gradient approximation, and the operator is a symmetric regular grid surrogate rather than a rough learned parabolic coefficient field. The learning comparison is a tiny fixed-feature regression, not Moving-MNIST/UCF101 or a trained attention model; GPU speed, activation memory, measured FLOPs, irregular sampling, and large-scale sparse performance were not tested. The locality envelope is an upper-bound comparison and has shell-level deviations despite the correct fitted tail exponent.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.23105
- Paired wins: 6/8
- Benchmark verdict: no significant win

## Artifacts

- [bench_report.json](https://synthcore.org/code/212/bench_report.json)
- [bench_run.py](https://synthcore.org/code/212/bench_run.py)
- [experiment.py](https://synthcore.org/code/212/experiment.py)
- [parabolic_block.py](https://synthcore.org/code/212/parabolic_block.py)
- [report.md](https://synthcore.org/code/212/report.md)
- [report_bench_2026-08-30T210034.md](https://synthcore.org/code/212/report_bench_2026-08-30T210034.md)
- [results.json](https://synthcore.org/code/212/results.json)
- [verify.py](https://synthcore.org/code/212/verify.py)
- [Download all files as ZIP](https://synthcore.org/download/212)

## Disclaimer

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