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NPN: Non-Linear Projections of the Null-Space for Imaging Inverse Problems
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Proposes NPN, a regularizer that trains a neural network to predict low-dimensional projections of the sensing matrix's null-space and uses it as a prior in PnP, unrolling, DIP and diffusion solvers for linear imaging inverse problems.
Problem
Imaging inverse problems are ill-posed because the sensing matrix has a non-trivial null-space; existing learned priors act in the image domain and do not explicitly constrain null-space components.
Approach
Design a projection matrix S whose rows lie in Null(H) (via QR, Fourier complements, Radon complement angles, or Toeplitz complement), learn a network G mapping measurements y to Sx*, and add gamma||G(y)-S x||^2 as a regularizer in variational/PnP solvers; S can be jointly optimized with orthogonality penalties. Includes convergence-rate and regularization-bound theorems for PnP.
Key results
- On CS with gamma=0.1, NPN-PnP reaches 21.12 dB vs 20.04 dB for the PnP baseline (CIFAR-10).
- On MRI AF=4, NPN-Sparsity achieves 38.16 dB vs 36.86 dB for the sparsity baseline.
- Deep image prior with NPN regularization gives improvements of up to 5 dB in deblurring (sigma=4.0) per the paper.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add NPN regularization to PnP reconstruction | PnP · CIFAR-10 SPC | PSNR: 20.04 -> 21.12 | Helped |
| Add NPN regularization to sparsity reconstruction | Sparsity reconstruction · MRI (AF = 4) | PSNR: 36.86 -> 38.16 | Helped |
| Add NPN regularization to RED reconstruction | RED · Deblurring (σ=2) | PSNR: 32.84 -> 33.67 | Helped |
| Set projection ratio p/n to 0.1 (ablation) | NPN PnP · CIFAR-10 SPC | PSNR: 20.04 -> 21.12 | Helped |
| Vary projection ratio p/n from 0.1 to 0.9 (ablation) | NPN PnP · CIFAR-10 SPC | PSNR: 21.12, 21.07, 20.78, 20.09, 20.41 | Mixed |
| Vary projection ratio p/n from 0.1 to 0.9 (ablation) | NPN Unrolling · CIFAR-10 SPC | PSNR: 24.32 -> 28.53, 28.75, 27.64, 26.73, 29.90 | Helped |
| Set projection ratio p/n to 0.1 (ablation) | NPN PnP · STL10 SPC | PSNR: 20.09 -> 19.91 | Hurt |
| Set projection ratio p/n to 0.3 (ablation) | NPN PnP · STL10 SPC | PSNR: 20.09 -> 21.14 | Helped |
| Vary projection ratio p/n from 0.1 to 0.9 (ablation) | NPN PnP · STL10 SPC | PSNR: 19.91, 21.14, 20.77, 20.31, 21.02 | Mixed |
| Vary projection ratio p/n from 0.1 to 0.9 (ablation) | NPN Unrolling · STL10 SPC | PSNR: 18.35 -> 19.64, 20.23, 18.76, 18.45, 19.48 | Helped |
Surprising findings
- Joint optimization of S is reported as unnecessary for well-structured matrices (Fourier, Toeplitz), where analytic complements suffice.
Limitations
- Requires training a dedicated network per sensing configuration (H, S), adding an offline training step.
- Two-stage training for integration with unrolling networks rather than end-to-end.
- Method may fail for some S designs where nonlinear relation between Hx and Sx is weak.
Experimental setup
Datasets: CIFAR-10 STL10 fastMRI knee CelebA LoDoPaB-CT Places365
Benchmarks: CIFAR-10 STL10 fastMRI knee CelebA
Metrics: PSNR
Baselines: PnP-FISTA RED-FISTA Sparsity-based FISTA DNSN DDN-I DDN-C
Builds on: plug-and-play priors null-space networks deep decomposition networks FISTA deep image prior
Closest papers by meaning
- A Restoration Network as an Implicit Prior (ICLR 2024 · similarity 0.88)
- FiRe: Fixed-points of Restoration Priors for Solving Inverse Problems (CVPR 2025 · similarity 0.88)
- Untrained Neural Nets for Snapshot Compressive Imaging: Theory and Algorithms (NeurIPS 2024 · similarity 0.87)
- Adapt and Diffuse: Sample-adaptive Reconstruction via Latent Diffusion Models (ICML 2024 · similarity 0.86)
- Stochastic Deep Restoration Priors for Imaging Inverse Problems (ICML 2025 · similarity 0.86)
- Improving Diffusion Models for Inverse Problems Using Optimal Posterior Covariance (ICML 2024 · similarity 0.86)
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