Riesz Fractional Variation Regularizer

(1) Integration point: attach the regularizer to the output of a coordinate-conditioned MLP, temporal encoder, or scalar neural field evaluated on an ordered grid. Let z=f_theta(x_grid) have shape [batch, channels, length]. (2) Pseudocode: choose a multiscale disjoint partition P containing intervals of lengths 1, 2, 4, ..., or sample M random disjoint intervals; for each [l,r], compute osc=max(z[...,l:r+1])-min(z[...,l:r+1]), then accumulate ((osc/(r-l+1)alpha)p). Take the p-th root and optimize loss=task_loss+lambda*R. Exact max/min gives subgradients; for smoother training use tau*logsumexp(z/tau) for the maximum and -tau*logsumexp(-z/tau) for the minimum, gradually reducing tau. (3) The paper supplies the scale-normalized oscillation and admissible range of alpha; the neural implementation estimates the supremum using sampled partitions, so report the sampling scheme and do not call it exact variation unless all partitions are enumerated. Start with p=2 and alpha in {0, 0.25, 0.5}; choose lambda so the penalty is initially 1–10% of task loss. (4) First experiment: noisy 1D regression with a fixed-width coordinate MLP, comparing this penalty against first-difference total variation and squared derivative penalties. Measure test MSE, high-frequency perturbation robustness, multiscale variation, and gradient-norm variance. Success is lower oscillation and better noisy extrapolation at equal training error; failure is oversmoothing, visible as degraded performance on high-frequency target functions.