$\mathrm{L}^p$ bounds for parabolic Riesz transforms with rough coefficients: The case $1<p \leq 2$
arXiv:2607.05181
2026
Architecture
2 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper develops a constructive multiscale calculus for rough, time-dependent parabolic operators, where spatial and temporal locality are controlled by complementary geometries. Its transferable asset is the combination of localized functional-calculus filters, explicit scale-ratio decay, and off-diagonal bounds that remain valid with merely measurable coefficients. This suggests replacing unrestricted spatiotemporal mixing with stable learned diffusion filters whose receptive field and temporal memory are controlled analytically across scales. A strong first test is a video or spatiotemporal graph model comparing these filters against dense temporal attention at matched FLOPs.
Ideas from this paper
✗ Failed on benchmark
2026
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.
Useful7/10
Difficulty6/10
Novelty6/10
Unverified
2026
Add a learned Riesz-transform branch that extracts normalized spatial gradients after diffusion by a positive parabolic operator. The diffusion branch carries smooth semantic content, while the Riesz branch represents boundaries, motion changes, and graph discontinuities. Resolvent smoothing makes the derivative branch less sensitive to feature noise than directly applying a finite difference.
Useful6/10
Difficulty5/10
Novelty7/10