# Positive Curvature Mirror-Descent DEQ

- ID: 1817
- Canonical URL: https://synthcore.org/idea/1817/positive-curvature-mirror-descent-deq
- API JSON: https://synthcore.org/api/idea/1817.json
- API Markdown: https://synthcore.org/api/idea/1817.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2608.04944](https://arxiv.org/abs/2608.04944)
- Category: architecture
- Solves: stability, accuracy, memory
- ML areas: cnn, optimizer, training-dynamics
- Math tags: geometry, optimization, probability, dynamical-systems
- Ratings: usefulness 8/10; difficulty 6/10; novelty 6/10

## 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.

## 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^*).

## Key formulas

- $$y=(Ax)\eta,\qquad \eta_i\sim\operatorname{Gamma}(L,1/L),\qquad \mathbb{E}[\eta_i]=1,\quad \operatorname{Var}[\eta_i]=1/L.$$
- $$D_L(y,Ax)=L\sum_{i=1}^{n}\left[\log (Ax)_i+\frac{y_i}{(Ax)_i}\right],\qquad \nabla_xD_L(y,Ax)=L A^{\mathsf T}\left(\frac{1}{Ax}-\frac{y}{(Ax)^2}\right).$$
- $$q_p(x)=\sqrt{1+\|\nabla x_p\|^2},\quad S(x)=\sum_pq_p(x),\quad H_p(x)=\nabla\!\cdot\left(\frac{\nabla x_p}{q_p(x)}\right),\quad R_{\alpha,\beta}(x)=\alpha S(x)+\frac{\beta}{2}\sum_pH_p(x)^2.$$
- $$x^{k+1}=x^k\odot\exp\left[-\tau\left(LA^{\mathsf T}\left(\frac{1}{Ax^k}-\frac{y}{(Ax^k)^2}\right)+\nabla R_{\alpha,\beta}(x^k)\right)\right],\qquad x^*=T_\theta(x^*).$$

## 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

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: harms
- Verdict: 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.

### Mechanism check

- Verdict: 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.
- Confidence: 8/10
- 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.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.0081
- Paired wins: 0/8
- Benchmark verdict: idea worse (significant)

## Artifacts

- [bench_report.json](https://synthcore.org/code/652/bench_report.json)
- [experiment.py](https://synthcore.org/code/652/experiment.py)
- [gamma_image_track.py](https://synthcore.org/code/652/gamma_image_track.py)
- [report.md](https://synthcore.org/code/652/report.md)
- [report_bench_2026-08-31T210442.md](https://synthcore.org/code/652/report_bench_2026-08-31T210442.md)
- [results.json](https://synthcore.org/code/652/results.json)
- [run_bench.py](https://synthcore.org/code/652/run_bench.py)
- [run_registered_bench.py](https://synthcore.org/code/652/run_registered_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/652)

## Disclaimer

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