✗ Failed on benchmark 2026

Two-Scale Parabolic Filter Block

Implementation & benchmark of arXiv:2607.05181 — $\mathrm{L}^p$ bounds for parabolic Riesz transforms with rough coefficients: The case $1<p \leq 2$

Usefulness7/10

Difficulty6/10

Novelty6/10

Architecture Stability Scalability Memory Ssm Attention Graph nn Memory Pde Harmonic analysis Functional analysis Geometry

Source paper: $\mathrm{L}^p$ bounds for parabolic Riesz transforms with rough coefficients: The case $1<p \leq 2$ arXiv:2607.05181 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

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.

Formulas

$$\mathcal{H}u=\partial_tu-\operatorname{Div}_x(A(x,t)\nabla_xu)$$

$$\psi(z)=z^{3\alpha}(1+z)^{-6\alpha},\qquad \varphi(z)=\bigl(1-(1+z)^{-\beta}\bigr)^{3\alpha}$$

$$\left\|\mathbf{1}_{F}\psi(\lambda^{2}\mathcal{H})\varphi(r^{2}\mathcal{H})\mathbf{1}_{E}u\right\|_{p}\leq C\left(1+\frac{d(E,F)}{\min(\lambda,r)}\right)^{-6\alpha}\left\|\mathbf{1}_{E}u\right\|_{p}$$

$$\left\|\psi(\lambda^{2}\mathcal{H})\varphi(r^{2}\mathcal{H})u\right\|_{p}\leq C\min\left(1,\left(\frac{r}{\lambda}\right)^{2\alpha}\right)\|u\|_{p}$$

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.

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

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: Inconclusive

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 8/10

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.

Agent confidence: 8/10

Baseline: Local one-hop smoother: test MSE 0.001317; FLOPs proxy 1,152
Idea: Two-scale parabolic features: test MSE 0.001162; FLOPs proxy 8,192. Exact scale exponent observed 2.996 vs predicted 3.0; locality slope observed -3.095 vs predicted -3.0; max scale-bound ratio 0.0567; noisy-operator max spectral norm 0.0568; block output and gradients finite.

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.

How to run: python3 verify.py && python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.05181) Sep 2, 2026

Ahead of the baseline (−7.1%) but not statistically significant (p=0.2311, wins 6 / 8 paired seeds).

Benchmark: Sequence modeling
Model: transformer_tiny
Paired seeds: 8
Baseline mean: 0.4709
Idea mean: 0.4377
Effect (Δ): -0.0332 (−7.1%; negative = idea better)
Wins: 6 / 8 paired seeds
p-value: 0.2311 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±13.0%
Mechanism: Not confirmed ✗
Practical effect: Inconclusive
Baseline tuning: swept over 3 configs

Limitations:

Only the registered built-in sequence track was tested. The implementation uses a fixed one-dimensional temporal Laplacian and dense eigendecomposition, not a learned rough spatiotemporal coefficient field or sparse Chebyshev/CG approximation. No vision, dynamics, tabular, irregular-sampling, activation-memory, or measured FLOP/speed comparison was run.

How to run: python3 bench_run.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Aug 30, 2026 latest Failed ✗

Agent confidence: 9/10

Built and evaluated a two-scale parabolic filter sequence model against the standard transformer_tiny on the structurally matched registered sequence track. The idea had lower mean MSE (0.437695 versus 0.470918; paired delta -0.033223; 6/8 wins), but the permutation p-value was 0.23105, so the improvement was not significant. The trained-model sensitivity signature also failed to confirm the predicted decay, observing -0.469 versus -3.0.

Baseline: 8-seed mean test MSE 0.470918; best baseline lr=0.001, weight_decay=0.0; baseline sweep means at lr 0.001/0.003/0.01: 0.479578/0.626773/1.033657.
Idea: 8-seed mean test MSE 0.437695; selected lr=0.001, weight_decay=0.0; idea sweep means at lr 0.001/0.003/0.01: 0.454809/0.456754/0.649332; paired delta=-0.033223; p=0.23105.

Benchmark result