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What's in a Prior? Learned Proximal Networks for Inverse Problems
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Proposes learned proximal networks (LPN), ICNN-gradient architectures that exactly implement proximal operators of learned nonconvex regularizers, trained with a proximal matching loss to recover the log-prior, and applies them in PnP-ADMM for deblurring, CT and compressed sensing.
Problem
Plug-and-play and unrolling methods use deep denoisers as proximal operators without guarantees that the networks are proximal, and the learned regularizer is uncharacterized; it is unclear whether the implicit prior matches the true data log-prior.
Approach
Parameterize the proximal operator as the gradient of an input-convex neural network (softplus activations, nonnegative weights, quadratic term for strong convexity), so it is exactly a proximal map for some nonconvex regularizer, and train it with a proximal matching loss (a Gaussian-kernel MAP-seeking loss in the small-gamma limit) that provably recovers the log-prior of the data distribution fr
Key results
- On Mayo-CT compressed sensing at compression rate 1/16, LPN reaches 38.03 PSNR / .919 SSIM versus 29.71 for AR and 26.54 for wavelet sparsity.
- On Mayo-CT sparse-view CT, LPN achieves 34.14 PSNR vs 33.48 for operator-agnostic AR and 34.76 for operator-specific UAR.
- On CelebA deblurring, LPN matches the best PnP-GS at sigma_blur=1, sigma_noise=.02 (33.0 PSNR, .92 SSIM) and is competitive across other settings.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use proximal matching instead of squared ℓ2 loss (ablation) | LPN trained on Laplacian samples · 1-D Laplacian denoising | match to ground-truth proximal | Helped |
| Use proximal matching instead of ℓ1 loss (ablation) | LPN trained on Laplacian samples · 1-D Laplacian denoising | match to ground-truth proximal | Helped |
| Use proximal matching instead of squared ℓ2 loss (ablation) | LPN trained on Laplacian samples · 1-D Laplacian prior estimation | match to ground-truth log-prior | Helped |
| Use proximal matching instead of ℓ1 loss (ablation) | LPN trained on Laplacian samples · 1-D Laplacian prior estimation | match to ground-truth log-prior | Helped |
| Use LPN instead of PnP-GS | PnP deblurring · CelebA deblurring | PSNR: 33.0 $\pm$ 3.0 -> 33.0 $\pm$ 2.9; 31.4 $\pm$ 2.4 -> 31.3 $\pm$ 2.3; 30.1 $\pm$ 2.5 -> 30.1 $\pm$ 2.4; 29.3 $\pm$ 2.3 -> | No clear effect |
| Use LPN instead of PnP-GS | PnP deblurring · CelebA deblurring | SSIM: .92 $\pm$ .03 -> .92 $\pm$ .03; .89 $\pm$ .03 -> .89 $\pm$ .03; .87 $\pm$ .04 -> .87 $\pm$ .04; .84 $\pm$ .05 -> .84 $\p | No clear effect |
| Use LPN instead of AR | Mayo-CT tomographic reconstruction · Mayo-CT tomographic reconstruction | PSNR: 33.48 -> 34.14 | Helped |
| Use LPN instead of AR | Mayo-CT tomographic reconstruction · Mayo-CT tomographic reconstruction | SSIM: .890 -> .891 | Helped |
| Use LPN instead of UAR | Mayo-CT tomographic reconstruction · Mayo-CT tomographic reconstruction | PSNR: 34.76 -> 34.14 | Hurt |
| Use LPN instead of UAR | Mayo-CT tomographic reconstruction · Mayo-CT tomographic reconstruction | SSIM: .897 -> .891 | Hurt |
| Use LPN instead of AR | Mayo-CT compressed sensing · Mayo-CT compressed sensing (compression rate =1/16) | PSNR: 29.71 -> 38.03 | Helped |
| Use LPN instead of AR | Mayo-CT compressed sensing · Mayo-CT compressed sensing (compression rate =1/4) | PSNR: 37.94 -> 44.05 | Helped |
Surprising findings
- Standard l2 and l1 losses learn posterior mean and median rather than the mode, so the learned prox deviates from soft-thresholding, whereas proximal matching recovers it.
Limitations
- Inverting the LPN to evaluate the regularizer at arbitrary points requires solving an optimization problem, and the gradient-based inversion is nonconvex without global guarantees.
- Proximal matching is applied with a finite gamma schedule, so the exact MAP guarantee holds only in the limit gamma to 0.
- Recovered prior is only identified up to an additive offset due to scaling symmetry.
Experimental setup
Datasets: MNIST CelebA Mayo-CT
Benchmarks: CelebA deblurring Mayo-CT sparse-view CT Mayo-CT compressed sensing
Metrics: PSNR SSIM
Baselines: PnP-BM3D PnP-DnCNN PnP-GS (Prox-DRUNet) adversarial regularizer (AR) unrolled adversarial regularizer (UAR) FBP wavelet sparsity
Builds on: input convex neural networks plug-and-play priors PnP-ADMM gribonval2020characterization proximal characterization
Closest papers by meaning
- Adaptive Proximal Gradient Methods for Structured Neural Networks (NeurIPS 2021 · similarity 0.90)
- Learning Proximal Operators to Discover Multiple Optima (ICLR 2023 · similarity 0.88)
- Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation (ICML 2024 · similarity 0.87)
- Proximal Algorithm Unrolling: Flexible and Efficient Reconstruction Networks for Single-Pixel Imaging (CVPR 2025 · similarity 0.87)
- MAP Estimation with Denoisers: Convergence Rates and Guarantees (NeurIPS 2025 · similarity 0.87)
- A Restoration Network as an Implicit Prior (ICLR 2024 · similarity 0.86)
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