Fourier-Calibrated Nonlocal Feature Gradient

1. Integration point: insert the module after an intermediate CNN feature tensor \(Z\in\mathbb{R}^{B\times C\times H\times W}\), before the next residual block. Use a local depthwise convolution as the baseline branch and add a nonlocal-gradient branch. For each spatial pixel \(p\), precompute offsets \(q\) inside radius \(\delta\), radial weights \(w_{q}=\exp(-\|q\|^{2}/(2\sigma^{2}))\), and normalize them so \(\sum_q w_q=1\). 2. Pseudocode: Y=0; G=0; for q in neighborhood(delta): D=shift(Z,q)-Z; G += w[q]*D; Y += Conv_q(D); Y=MLP(concat(Z,G))+Z; F=FFT2(Y); penalty=sum_c,sum_k m[k]*abs(F[c,k])**2/(H*W); loss=task_loss+lambda_spec*penalty. Here \(G\) is the discrete version of \(\mathcal{G}_{\delta}\), and \(m[k]\) is computed from the discrete multiplier formula: m[k]=sum_q w[q]*(1-cos(2*pi*dot(k,q))). Use a real FFT and include only nonredundant frequencies. 3. The paper supplies the nonlocal difference construction, the positive \(1-\cos(\cdot)\) multiplier, and the local-consistency rationale. Compute \(m\) numerically from the chosen kernel rather than assuming a continuum approximation. Keep \(\delta\) and \(\lambda_{spec}\) fixed initially; in a later ablation, learn \(\sigma\) using a softplus parameterization and renormalize weights at every forward pass. 4. First cheap experiment: train ResNet-18 on CIFAR-10 and compare the baseline, a standard dilated convolution with the same receptive field, and this module at matched parameter count and FLOPs. Measure clean accuracy, CIFAR-10-C corruption accuracy, gradient-norm variance, feature-map high-frequency energy, and wall-clock overhead. Success means equal-or-better clean accuracy, lower high-frequency energy and gradient variance, and improved corruption accuracy with no more than 10% training overhead. Ablate the nonlocal branch and multiplier penalty separately, and test \(\delta\in\{1,2,4\}\) pixels to verify whether shrinking \(\delta\) approaches the local baseline.