Circular Morera Regularizer
Implementation & benchmark of arXiv:2608.04540 — An Infinitesimal Circular Morera Theorem
Source paper: An Infinitesimal Circular Morera Theorem arXiv:2608.04540 ⓘ · analyzed Aug 31, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Add a multiscale circular-integral penalty to a complex-valued neural field f_theta: R^2 -> C. The penalty directly tests the local contour condition that characterizes holomorphic functions, providing a derivative-free alternative to explicitly penalizing the Cauchy-Riemann residual.
Formulas
Mathematical statement
The paper defines partial = 1/2(partial_x - i partial_y) and bar-partial = 1/2(partial_x + i partial_y), and proves that if D is a domain, f is continuous on D, and for every center a in D, the contour integral of f over the circle of radius r centered at a is o(r^2) as r tends to zero, then f is holomorphic, equivalently bar-partial f = 0. Here a is the complex circle center, r is the radius, and o(r^2) means that the magnitude of the integral divided by r^2 tends to zero. For a smooth field, the local expansion is integral over the circle of f(zeta) dzeta = 2 pi i r^2 bar-partial f(a) + O(r^3), so the normalized circular residual estimates the antiholomorphic derivative without automatic differentiation. The adapted loss is L_circ = E over centers a and radii r of the squared magnitude of the normalized contour integral, with the contour integral approximated by quadrature.
Implementation notes
1. Integration point: use a complex-valued MLP f_theta(x,y) = u_theta(x,y) + i v_theta(x,y) as a coordinate-field decoder, image representation, or analytic latent embedding. Add lambda times L_circ to the reconstruction or supervised loss during training; inference is unchanged. Sample centers a = (x,y) from training coordinates and reject circles that leave the valid domain. 2. Pseudocode: for each batch of centers a: sample radii r from logarithmically spaced values; sample t_j = 2*pi*j/K for j=0,...,K-1; z_j = a + r*exp(i*t_j); f_j = f_theta(z_j); dz_j = i*r*exp(i*t_j)*(2*pi/K); I = sum_j f_j*dz_j; q = I/(2*pi*i*r*r); L_circ += abs(q)^2; loss = L_task + lambda*mean(L_circ); backprop(loss). Start with K = 16 or 32 and two to four radii. Gradients must flow through every f_theta(z_j). 3. The theorem supplies the o(r^2) certificate and the 2*pi*i*r^2*bar-partial f(a) interpretation. Quadrature error, radius range, lambda, and the number of contour samples are estimated or selected empirically. Normalize by r^2 to prevent large circles from dominating, and optionally use the maximum residual across radii to prevent the model from satisfying only one scale. 4. First experiment: train a small complex MLP on analytic targets such as f(z) = z^3 + 0.2 z^2 and on nonanalytic targets containing a conjugate-z term. Compare ordinary MSE, MSE plus the circular penalty, and MSE plus an explicit Cauchy-Riemann penalty. Then evaluate a 64 by 64 coordinate decoder on a smooth image-fitting task. Measure task error, circular residual at unseen centers and radii, extrapolation outside the training region, and robustness to noisy coordinate samples. Success means lower unseen-region error or better noise robustness at matched parameter count, with normalized residual decreasing consistently as radii shrink.
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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