Elliptic Low-Oscillation Feature Layer

(1) Integration point: implement a 2-D or 3-D implicit feature block operating on a feature map $f\in\mathbb{R}^{C\times H\times W}$. A small CNN predicts one symmetric diffusion tensor $A_c(x)$ and drift vector $b_c(x)$ per channel or per channel group. Compute finite-difference approximations of $D_i u$ and $D_{ij}u$, then define $L_Au=A:D^2u+b\cdot\nabla u$. Replace an ordinary residual update by $u_{k+1}=u_k+\eta(f-u_k+\tau L_Au_k)$, or solve $(I-\eta\tau L_A)u=f$ with conjugate gradients or 10--20 unrolled Jacobi iterations. (2) Pseudocode: predict raw parameters $d_1,d_2,d_3$ and rotation angle $\theta$; form $Q(\theta)$ and $A=Q\,\mathrm{diag}(\lambda+(\Lambda-\lambda)\sigma(d_j))Q^T$, enforcing eigenvalues in $(\lambda,\Lambda)$; predict $b$; compute $A_P$ on sliding patches; add $\gamma\mathcal{R}_{\mathrm{BMO}}(A)$ to the task loss; apply the implicit solver and backpropagate through the unrolled iterations. (3) Ellipticity is guaranteed analytically by the parameterization. Estimate BMO empirically at patch radii $3\times3$, $7\times7$, and $15\times15$; use a soft maximum rather than the exact supremum. Estimate the drift control by $R^{1-n/q}(\frac{1}{|P|}\sum_{x\in P}|b(x)|^q)^{1/q}$. (4) First experiment: train on CIFAR-10 or ImageNet-32, replacing a depthwise convolution in a small ResNet. Compare unconstrained dynamic convolution, ellipticity-only, and ellipticity-plus-BMO variants at equal parameter count and FLOPs. Record training loss versus steps, gradient explosion frequency, solver residuals, corruption accuracy, and accuracy under perturbations of predicted coefficients. Success means smoother optimization, fewer solver failures, improved corruption robustness, or equal accuracy with fewer unrolled iterations.