Variable-Exponent Fourier Block
Implementation & benchmark of arXiv:2607.15922 — Fourier inequalities in variable Lebesgue spaces
Source paper: Fourier inequalities in variable Lebesgue spaces arXiv:2607.15922 ⓘ · analyzed Aug 30, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Replace a fixed-norm Fourier feature layer by a Fourier transform followed by spatially varying modular normalization. Use a baseline exponent approaching the endpoint regime at large coordinates and permit only bounded, smooth deviations so the transform remains controlled while the network can emphasize localized details.
Formulas
Mathematical statement
The paper defines the Fourier transform by $\hat f(\xi)=\int_{\mathbb R}f(x)e^{-2\pi ix\xi}\,dx$ and recalls Hausdorff--Young, $\|\hat f\|_{p'}\leq\|f\|_p$, where $1/p+1/p'=1$. Its variable-exponent construction uses $r(x)=\left(1-\frac{1/2}{\ln(e+|x|)}\right)^{-1}$, which satisfies $r(x)\to1$ as $|x|\to\infty$, together with the assumption $1/r(\cdot)\in\operatorname{LH}_{\infty}$, meaning the reciprocal exponent has controlled logarithmic variation at large distances. Under these conditions, $\mathscr F:L^{r(\cdot)}(\mathbb R)\to L^{r'(\cdot)}(\mathbb R)$ is bounded. The paper also allows a finite-measure perturbation: with $\Omega=\bigcup_{n\ge1}(n,n+2^{-n})$, $|\Omega|<\infty$, define $p(x)=2$ and $q(x)=3/2$ on $\Omega$, and $p(x)=r(x)$ and $q(x)=r'(x)$ outside $\Omega$; then $\mathscr F:L^{p(\cdot)}\to L^{q(\cdot)}$ remains bounded. For a tensor $z$, use the variable modular $\rho_p(z)=\int |z(x)|^{p(x)}dx$ and Luxemburg norm $\|z\|_{p(\cdot)}=\inf\{\lambda>0:\rho_p(z/\lambda)\le1\}$. The neural module exploits the bounded map between these nonuniform norms instead of assuming that a pointwise conjugate exponent alone guarantees stability.
Implementation notes
Integrate the block after a convolutional stem and before the next residual stage. Let the input be $X\in\mathbb R^{B\times C\times H\times W}$. Apply a channelwise 2D FFT, multiply coefficients by a learned complex diagonal gain $G_{c,u,v}$, and apply the inverse FFT. Predict a bounded exponent field with a small depthwise $3\times3$ convolution: $m=\operatorname{sigmoid}(g(X))$, then set $p=r_\alpha+(2-r_\alpha)m$, clipping $p$ to $[1.05,1.95]$, and set $q=p/(p-1)$. Here $r_\alpha(x)=\left(1-\frac{1/2}{\log(e+\alpha\|x\|)}\right)^{-1}$ is the finite-grid version of the paper's $r(x)$, with normalized image coordinate $x$ and fixed scale $\alpha$. Normalize the inverse-FFT tensor $Z$ using the Luxemburg norm: solve $\rho_p(Z/\lambda)=1$ by five bisection iterations, where the discrete modular is $\rho_p(Z/\lambda)=N^{-1}\sum_i(|Z_i|/\lambda)^{p_i}$. Pseudocode: $Y=\operatorname{FFT}(X)$; $Y\leftarrow G\odot Y$; $Z=\operatorname{IFFT}(Y)$; $p\leftarrow r_\alpha+(2-r_\alpha)\operatorname{sigmoid}(g(X))$; $q\leftarrow p/(p-1)$; solve $N^{-1}\sum_i(|Z_i|/\lambda)^{p_i}=1$; output $Z/\lambda$. Add $\mathcal L_{LH}=\sum_{(i,j)\in E}|p_i-p_j|/\log(e+1/d_{ij})$ to penalize rapid spatial exponent changes, plus a weak penalty keeping $p$ near $r_\alpha$. The paper supplies the exponent construction and boundedness motivation; estimate the empirical operator ratio $\hat C=\|\operatorname{FFT}(Z)\|_{q(\cdot)}/\|Z\|_{p(\cdot)}$ on minibatches to detect instability. First test on CIFAR-10 using a small ResNet-18, comparing this block against a standard learned FFT-gain layer with fixed $L^2$ normalization at equal parameter count and FLOPs. Success is higher validation accuracy or corruption robustness with bounded modular-norm ratios and no training instability, especially on localized-texture examples.
Verification
This idea has not been verified yet.
Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.
Artifacts
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