# Learned fractional-scale convolution

- ID: 1136
- Canonical URL: https://synthcore.org/idea/1136/learned-fractional-scale-convolution
- API JSON: https://synthcore.org/api/idea/1136.json
- API Markdown: https://synthcore.org/api/idea/1136.md
- Verification status: unverified
- Source: [arXiv:2607.19076](https://arxiv.org/abs/2607.19076)
- Category: architecture
- Solves: accuracy, stability, generalization
- ML areas: cnn, regularization
- Math tags: pde, harmonic-analysis, functional-analysis, measure-theory
- Ratings: usefulness 6/10; difficulty 5/10; novelty 7/10

## Idea description

Replace a single Laplacian or fixed diffusion regularizer in a CNN with a finite positive mixture of fractional Laplacians at several orders. The resulting module separately controls short-range smoothing and long-range spatial coupling, while positivity preserves a dissipative energy and avoids the unstable behavior of arbitrary signed mixtures.

## Mathematical statement

The paper defines the superposition operator \(\mathcal{L}_{\mu}u=\int_{[0,1]}(-\Delta)^s u\,d\mu(s)\), where \(s\in[0,1]\) is the fractional order and \(\mu=\mu^+-\mu^-\) is a signed Borel measure. For \(s\in(0,1)\), \((-Δ)^s u(x)=C_{N,s}\int_{\mathbb{R}^N}[2u(x)-u(x+y)-u(x-y)]/|y|^{N+2s}\,dy\), with \(C_{N,s}=2^{2s}s\Gamma((N+2s)/2)/(\pi^{N/2}\Gamma(1-s))\). Approximate the measure by \(\mu=\sum_{k=1}^K w_k\delta_{s_k}\), constrain \(w_k\ge0\) and \(\sum_k w_k=1\), and exploit the Fourier multiplier \(\widehat{(-\Delta)^s u}(\xi)=|\xi|^{2s}\hat u(\xi)\). The mixture therefore has nonnegative multiplier \(m(\xi)=\sum_k w_k|\xi|^{2s_k}\), and its quadratic energy \(R(u)=\frac12\sum_k w_k\|(-\Delta)^{s_k/2}u\|_2^2\) has gradient \(\sum_k w_k(-\Delta)^{s_k}u\).

## Key formulas

- $$\mathcal{L}_{\mu}u:=\int_{[0,1]}(-\Delta)^{s}u\,d\mu(s)$$
- $$(-\Delta)^{s}u(x):=C_{N,s}\int_{\mathbb{R}^{N}}\frac{2u(x)-u(x+y)-u(x-y)}{|y|^{N+2s}}\,dy,\qquad C_{N,s}=\frac{2^{2s}s\Gamma\left(\frac{N+2s}{2}\right)}{\pi^{N/2}\Gamma(1-s)}$$
- $$\mu=\sum_{k=1}^{K}w_k\delta_{s_k},\quad w_k=\frac{e^{a_k}}{\sum_{j=1}^{K}e^{a_j}},\quad \widehat{\mathcal{L}_{\mu}u}(\xi)=\left(\sum_{k=1}^{K}w_k|\xi|^{2s_k}\right)\hat u(\xi)$$
- $$R(u)=\frac12\sum_{k=1}^{K}w_k\|(-\Delta)^{s_k/2}u\|_2^2,\qquad \nabla_u R(u)=\sum_{k=1}^{K}w_k(-\Delta)^{s_k}u$$

## Implementation notes

(1) Integration point: add this as a residual spatial module after a CNN feature block, or as a feature regularizer before normalization. For an input tensor \(u\in\mathbb{R}^{B\times C\times H\times W}\), choose fixed orders \(s_k\in\{0.1,0.25,0.5,0.75,0.95\}\), shared across channels initially, and learn logits \(a_k\). Use periodic padding in the first prototype so the fractional operators can be evaluated exactly in the Fourier domain. Apply a 2-D FFT independently to every batch/channel plane, multiply by \(m(\xi)=\sum_k\operatorname{softmax}(a)_k(|\xi_1|^2+|\xi_2|^2+\varepsilon)^{s_k}\), and inverse transform. Return \(y=u-\tau\mathcal{L}_\mu u\) as a residual diffusion layer, with \(\tau=\operatorname{softplus}(b)\) initialized near 0.1; alternatively add \(\lambda R(u)\) to the task loss. (2) Pseudocode: `w = softmax(a); freq2 = fx**2 + fy**2; m = sum(w[k] * (freq2 + eps)**s[k] for k); Lu = ifft2(m * fft2(u)).real; y = u - tau * Lu`; for the loss version, compute `R = 0.5 * sum(w[k] * mean(abs(ifft2((freq2 + eps)**(s[k]/2) * fft2(u)))**2) for k)` and add `lambda * R` to the task loss. (3) The fractional orders, Fourier multiplier, and positivity constraint come from the paper's superposition construction; \(a_k\), \(\tau\), and \(\lambda\) are learned parameters. No theorem constants need to be estimated; monitor the learned weights and feature spectra empirically. (4) First cheap experiment: train a small CIFAR-10 ResNet-18 with identical parameter count, augmentation, optimizer, and training budget, comparing the baseline, fixed integer-Laplacian smoothing, and the learned fractional mixture. Use three seeds. Success means higher validation accuracy or corruption robustness at equal FLOPs, stable loss curves, and nontrivial specialization of weights across layers. Also test repeated application of the module and verify that feature energy does not grow unexpectedly.

## Disclaimer

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