△ Mechanism confirmed, baseline not beaten 2026

Positive Curvature Mirror-Descent DEQ

Implementation & benchmark of arXiv:2608.04944 — A geometry-based deep equilibrium model for image restoration under multiplicative Gamma noise

Usefulness8/10

Difficulty6/10

Novelty6/10

Architecture Stability Accuracy Memory Cnn Optimizer Training dynamics Geometry Optimization Probability Dynamical systems

Source paper: A geometry-based deep equilibrium model for image restoration under multiplicative Gamma noise arXiv:2608.04944 ⓘ · analyzed Aug 31, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

Idea description

Replace the black-box equilibrium denoiser in an image-restoration DEQ with a positivity-preserving mirror-descent equilibrium driven by the exact Gamma likelihood and a discretized surface-area/mean-curvature regularizer. The equilibrium layer has a small number of learned scalar or channel-wise parameters instead of a large implicit CNN, while the exponentiated update prevents negative intensities and naturally matches multiplicative noise.

Formulas

$$y=(Ax)\eta,\qquad \eta_i\sim\operatorname{Gamma}(L,1/L),\qquad \mathbb{E}[\eta_i]=1,\quad \operatorname{Var}[\eta_i]=1/L.$$

$$D_L(y,Ax)=L\sum_{i=1}^{n}\left[\log (Ax)_i+\frac{y_i}{(Ax)_i}\right],\qquad \nabla_xD_L(y,Ax)=L A^{\mathsf T}\left(\frac{1}{Ax}-\frac{y}{(Ax)^2}\right).$$

$$q_p(x)=\sqrt{1+\|\nabla x_p\|^2},\quad S(x)=\sum_pq_p(x),\quad H_p(x)=\nabla\!\cdot\left(\frac{\nabla x_p}{q_p(x)}\right),\quad R_{\alpha,\beta}(x)=\alpha S(x)+\frac{\beta}{2}\sum_pH_p(x)^2.$$

$$x^{k+1}=x^k\odot\exp\left[-\tau\left(LA^{\mathsf T}\left(\frac{1}{Ax^k}-\frac{y}{(Ax^k)^2}\right)+\nabla R_{\alpha,\beta}(x^k)\right)\right],\qquad x^*=T_\theta(x^*).$$

Mathematical statement

The observation model in the paper is y=(Ax)\eta, where x\in\mathbb{R}_{>0}^n is the clean image, A is a known blur or measurement operator, y is observed data, and \eta has independent Gamma components with mean 1 and variance 1/L. Taking \eta_i\sim\mathrm{Gamma}(L,1/L), the negative log-likelihood for z=Ax, up to terms independent of z, is D_L(y,z)=L\sum_i[\log z_i+y_i/z_i]. Its gradient is
abla_zD_L=L(1/z-y/z^2), and
abla_xD_L=A^T
abla_zD_L. For an image viewed as a graph, define q(x)=\sqrt{1+\|
abla x\|^2}, the surface-area functional S(x)=\sum_p q_p(x), and the mean curvature H(x)=
abla\cdot(
abla x/q(x)); use the explicit regularizer R_{\alpha,\beta}(x)=\alpha S(x)+(\beta/2)\sum_p H_p(x)^2, with \alpha,\beta\ge0. The total objective is F(x)=D_L(y,Ax)+R_{\alpha,\beta}(x). Entropic mirror descent uses h(x)=\sum_i(x_i\log x_i-x_i), B_h(u,v)=\sum_i[u_i\log(u_i/v_i)-u_i+v_i], and produces the positivity-preserving update x^{k+1}=x^k\odot\exp[- au
abla F(x^k)]. A damped DEQ map is T_ heta(x)=\exp((1-\lambda)\log x+\lambda[\log x- au
abla F_ heta(x)]), with \lambda\in(0,1], and the equilibrium satisfies x^*=T_ heta(x^*).

Implementation notes

(1) Integration point: implement a restoration block taking a positive observation y and known nonnegative blur A. Parameterize x=softplus(u)+epsilon only for initialization, then maintain x directly in log-space inside the equilibrium solver. The trainable parameters should initially be log_tau, logit_lambda, log_alpha, and log_beta; optionally make alpha and beta channel-wise. Compute A x using the known convolution and A^T using its adjoint. Compute forward and backward finite differences with reflective boundaries. Build q=sqrt(1+dx^2+dy^2), surface area S=sum(q), curvature H=div([dx,dy]/q), and obtain the gradient of R through automatic differentiation. (2) Pseudocode: initialize x0=clamp(y,eps,infty); for k in 1..T, z=A(x); g_data=L*A_transpose(1/(z+eps)-y/(z+eps)^2); q=sqrt(1+grad(x)^2); H=div(grad(x)/q); R=alpha*sum(q)+0.5*beta*sum(H^2); g=g_data+autograd(R,x); x=exp((1-lambda)*log(x)+lambda*(log(x)-tau*g)); stop when relative change is below tolerance. At inference, use Anderson acceleration only after verifying that the raw map is stable. (3) The Gamma fidelity, mirror update, positivity invariant, and curvature definitions come from the mathematical construction; alpha, beta, tau, lambda, the iteration count, and finite-difference discretization are estimated or learned empirically. (4) First experiment: train on 128x128 synthetic grayscale images blurred by a 9x9 Gaussian kernel and corrupted with L in {1,4} Gamma noise. Compare against a standard 6- to 12-block CNN-regularized DEQ and an unrolled Adam/gradient-descent solver at equal forward evaluations. Measure PSNR/SSIM, negative-pixel rate, equilibrium residual, peak memory, and trainable parameter count. Success means equal-or-better PSNR with at least 10x fewer learned parameters, no positivity violations, and a lower or more predictable residual curve as T increases. Also ablate surface area and curvature separately to test whether the geometric terms—not merely the Gamma likelihood—provide the gain.

Verification

Mechanism confirmed, baseline not beaten

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Confirmed ✓

Stage 2 · Practical benchmark result: Significantly worse ✗

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 8/10

Built a positivity-preserving Gamma-likelihood mirror-descent restoration solver with differentiable surface-area and mean-curvature regularizers, plus a projected-GD baseline. The core mechanism manifested: measured local contraction ratios closely matched |1−tau·L/y|, with the predicted stability transition at tau·L/y=2, and all 100 exponentiated iterates remained strictly positive. The restoration toy did not beat projected GD in PSNR, although curvature produced a small consistent improvement over likelihood-only mirror descent.

Agent confidence: 8/10

Baseline: Projected GD: 11.76 dB (L=1), 17.13 dB (L=4); final relative residuals 0.00167 and 0.00327; minimum value 1e-6.
Idea: Mirror descent with Gamma data term: 10.92 dB (L=1), 14.77 dB (L=4); final relative residuals 0.00497 and 0.00727; strictly positive minima 2.20e-4 and 1.23e-3. Adding surface area plus curvature increased PSNR slightly to 10.94/14.79 dB and lowered residuals marginally. Gamma gradient finite-difference max error was 7.1e-8. Stability observed/predicted ratios at tau·L/y={0.25,0.75,1.5,1.9,2.1} were {0.752/0.75,0.253/0.25,0.494/0.50,0.871/0.90,1.058/1.10}.
Parameter count: No learned parameters in this fixed-parameter MVP.

Limitations: This is a small 32x32 synthetic grayscale experiment, not the proposed 128x128 trained DEQ. Parameters were fixed rather than learned, no CNN-regularized DEQ or SSIM/memory/FLOP comparison was run, and the finite-difference implementation uses periodic boundaries rather than reflective boundaries. The stability prediction was verified for the scalar A=I case, not for the full blurred nonlinear restoration map.

How to run: python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.04944) Sep 2, 2026

Significantly WORSE than the tuned baseline (+1573.9%, p=0.0081, wins 0 / 8 paired seeds).

Benchmark: Tabular regression (Friedman)
Model: mlp_tiny
Paired seeds: 8
Baseline mean: 13.957
Idea mean: 233.6266
Effect (Δ): 219.6696 (+1573.9%; negative = idea better)
Wins: 0 / 8 paired seeds
p-value: 0.0081 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±436.5%
Mechanism: Confirmed ✓
Practical effect: Hurts
Baseline tuning: swept over 8 configs

Limitations:

The registered tabular track tests the optimizer/regularizer aspect rather than the full image-restoration DEQ. CNN DEQ restoration, image PSNR/SSIM, large images, Anderson acceleration, memory, FLOPs, and the full proposed learned equilibrium layer were not tested.

How to run: python3 run_registered_bench.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Aug 31, 2026 latest Failed ✗

Agent confidence: 10/10

The optimizer/regularizer aspect was tested on the structurally appropriate registered tabular track using identical mlp_tiny systems. Baseline MSE was 13.9569603 and the exponentiated mirror-update idea reached 233.6265755; paired delta was +219.6696151 with permutation p=0.0081 and zero idea wins. Positive parameter magnitudes were preserved, but the independent task metric showed a significant loss.

Baseline: Mean MSE 13.9569603, std 1.7910383; best lr=0.003, wd=0.0.
Idea: Mean MSE 233.6265755, std 68.5658324; best lr=0.003, wd=0.0, lambda=0.5; 0/8 wins.

Benchmark result