GSNR: Graph Smooth Null-Space Representation for Inverse Problems

Romario Gualdrón-Hurtado; Roman Jacome; Rafael S. Suarez; Henry Arguello

cs.CVFeb 23, 2026

GSNR: Graph Smooth Null-Space Representation for Inverse Problems

Authors: Romario Gualdrón-Hurtado, Roman Jacome, Rafael S. Suarez, Henry Arguello

Organizations: Universidad Industrial de Santander, Colombia, 680002

Abstract

Inverse problems in imaging are ill-posed, leading to infinitely many solutions consistent with the measurements due to the non-trivial null-space of the sensing matrix. Common image priors promote solutions on the general image manifold, such as sparsity, smoothness, or score function. However, as these priors do not constrain the null-space component, they can bias the reconstruction. Thus, we aim to incorporate meaningful null-space information in the reconstruction framework. Inspired by smooth image representation on graphs, we propose Graph-Smooth Null-Space Representation (GSNR), a mechanism that imposes structure only into the invisible component. Particularly, given a graph Laplacian, we construct a null-restricted Laplacian that encodes similarity between neighboring pixels in the null-space signal, and we design a low-dimensional projection matrix from the $p$ -smoothest spectral graph modes (lowest graph frequencies). This approach has strong theoretical and practical implications: i) improved convergence via a null-only graph regularizer, ii) better coverage, how much null-space variance is captured by $p$ modes, and iii) high predictability, how well these modes can be inferred from the measurements. GSNR is incorporated into well-known inverse problem solvers, e.g., PnP, DIP, and diffusion solvers, in four scenarios: image deblurring, compressed sensing, demosaicing, and image super-resolution, providing consistent improvement of up to 4.3 dB over baseline formulations and up to 1 dB compared with end-to-end learned models in terms of PSNR.

Figures & tables

Figure 1: For image SR task with $SRF=4,n=3\cdot 128^{2}$ , we show ground-truth, adjoint reconstruction, NS projection, projection onto graph-smooth NS with $\mathbf{L}_{4nn}$ and $\mathbf{L}_{8nn}$ .

Figure 2: Coverage and spectral analysis for SR task. (a) $\mathbf{x^{\ast}}=\mathbf{H}^{\top}\mathbf{Hx^{\ast}+S}^{\top}\mathbf{Sx^{\ast}}$ for $\mathbf{S}$ given by ( 6 ) using $\mathbf{L}=\mathbf{I}$ (orange), $\mathbf{L=L}_{4nn}$ (green) and $\mathbf{L=L}_{8nn}$ (blue) for different values of $p$ . (b) RNSD representation error, varying $p$ . (c) Variation of $\mathbf{T}$ normalized eigenvalues with respect to their index.

Figure 3: Per-mode predictability for each case when $\mathbf{L}\in\{\mathbf{I},\mathbf{L}_{8nn},\mathbf{L}_{4nn}\}$ for SR case with $SRF=4$ with $n=64^{2}$ .

Figure 4: a) Coverage and b) Predictability vs.

p

for CS with CIFAR-10 dataset. In this case,

m/n=0.1

and

p=1\cdots n-m

Figure 5: Results of DM-based solvers (DPS [ 8 ] & DiffPIR [ 55 ] ) for Baseline, NPN [ 17 ] , and GSNR with $\mathbf{L_{4nn}}$ and $\mathbf{L_{8nn}}$ . Here, $p=0.1n$ .

Method	Task
Method	Super-Resolution	Demosaicing
PnP [ 20 ]	27.37	39.35
NPN [ 17 ]	29.21	39.77
DDN-Cascade [ 7 ]	28.92	37.83
DDN-Independent [ 7 ]	28.81	38.92
DNSN [ 41 ]	26.52	39.33

Table 1: Quantitative comparison on the super-resolution and demosaicing tasks (higher is better) using the CelebA test set. For each task, the best result and second-best results.

Method	$\gamma_{\mathrm{g}}$	Places365		CelebA
Method	$\gamma_{\mathrm{g}}$	Lip-DnCNN	Wavelet	Lip-DnCNN	Wavelet
Baseline (PGD-PnP)	–	31.58	30.78	35.26	33.64
NPN [ 17 ]	–	33.22	33.17	37.86	37.68
GSNR w. $\mathbf{L}_{8\text{nn}}$	✘	33.60	33.59	38.18	37.99
GSNR w. $\mathbf{L}_{8\text{nn}}$	✓	33.60	33.59	38.18	37.99

Table 2: Final PSNR (dB) for deblurring (

\sigma_{k}=1

) with

n=64^{2}\cdot 3

and

p=0.8\cdot n

. We report results on Places365 [ 11 ] and CelebA [ 25 ] test sets using Lip-DnCNN [ 40 ] and a wavelet denoiser [ 12 ] . For each column, best and second-best results.

Figure 6: Deblurring convergence curves for PnP-PGD baseline using Lip-DnCNN and GSNR variants with

\mathbf{L}_{8nn}

with and without the null-restricted Laplacian regularization.

Figure 7: Results of PnP, NPN, and GSNR varying $\mathbf{L}$ and denoisers. Best results for each denoiser are in bold. Here, $p=0.1n$ .

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 8: SR with different graph Laplacians. Left: ground truth $\mathbf{x}^{\ast}$ , its null component $\mathbf{P}_{n}\mathbf{x}^{\ast}$ , and graph-smoothed null responses $\mathbf{P}_{n}\mathbf{L}_{\mathrm{sym}}\mathbf{P}_{n}\mathbf{x}^{\ast}$ and $\mathbf{P}_{n}\mathbf{L}_{\mathrm{rw}}\mathbf{P}_{n}\mathbf{x}^{\ast}$ . Right: PSNR vs. iteration for GSNR-PnP with $\mathbf{L}\in\{\mathbf{L}_{\mathrm{sym}},\mathbf{L}_{\mathrm{rw}},\mathbf{I}\}$ , and the PnP baseline.

Figure 9: (a) Variation of

\mathbf{T}

normalized eigenvalues for

\mathbf{L}_{\theta}

with respect to their index in CS. (b) Coverage of

\mathbf{S}

with

\mathbf{L}_{\theta}

in CS.

	$\displaystyle\mathbf{x}_{i}\leftarrow\mathbf{x}_{i-1}-\alpha$	$\displaystyle\left(\mathbf{H}^{\top}(\mathbf{Hx}_{i-1}-\mathbf{y})+{\color[rgb]{0,0.6,0}\gamma\mathbf{S}^{\top}(\mathbf{Sx}_{i-1}-\mathrm{G}^{*}(\mathbf{y}))}\right.$
		$\displaystyle\left.+{\color[rgb]{0,0,1}\gamma_{g}\mathbf{Tx}_{i-1}}\right)$

Appendix

Algorithm 2 GSNR PnP-PGD with null-space and graph regularization

Figure 10: PnP variant with null-only projector regularizer: illustrative convergence.

Figure 12: Effect of the null-only graph regularizer on GSNR-PnP convergence for demosaicing. We plot PSNR versus iteration for GSNR-PnP with different Laplacians (

\mathbf{L}=\mathbf{I}

\mathbf{L}_{4\mathrm{nn}}

\mathbf{L}_{8\mathrm{nn}}

) and with the graph-regularization weight

\gamma_{\mathrm{g}}

either enabled or disabled, along with the standard PnP baseline (dashed).

Figure 13: Effect of the null-only graph regularizer on GSNR-PnP convergence for super-resolution. We plot PSNR versus iteration for GSNR-PnP with different Laplacians (

\mathbf{L}=\mathbf{I}

\mathbf{L}_{4\mathrm{nn}}

\mathbf{L}_{8\mathrm{nn}}

) and with the graph-regularization weight

\gamma_{\mathrm{g}}

either enabled or disabled, along with the standard PnP baseline (dashed).

Figure 16: Spectral coverage curves

C(p)

for super-resolution, comparing different Laplacian choices in the GSNR construction: grid

\mathbf{L}_{4\mathrm{nn}}

, grid

\mathbf{L}_{8\mathrm{nn}}

, random-walk normalized

\mathbf{L}_{\mathrm{rw}}

, symmetric normalized

\mathbf{L}_{\mathrm{sym}}

, and the geometry-free baseline

\mathbf{L}=\mathbf{I}

. Coverage

C(p)

is the fraction of null-space variance captured by the first

p

graph-smooth modes.

Figure 17: Results using latent-space diffusion models for SR.

Figure 18: Convergence comparison with PnP, NPN, and GSNR for CS.

Parameter	Value
$\gamma_{\mathrm{g}}$	0.01
$\gamma$	0.1
$\eta$	0.01
$\alpha$	$\frac{1.0}{\\|\mathbf{H}\\|}$
$K$ (Max iterations)	3000

Appendix

Table 3: Experimental settings for GSNR PGD-PnP in demosacing.

Method	$\gamma_{\mathrm{g}}$	Lip-DnCNN [ 40 ]	DRUNet [ 52 ]
Baseline	–	39.35	27.91
NPN [ 17 ]	✘	39.77	30.12
GSNR w. $\mathbf{L}_{4nn}$	✘	39.79	30.14
GSNR w. $\mathbf{L}_{4nn}$	✓	39.89	30.13
GSNR w. $\mathbf{L}_{8nn}$	✘	39.77	30.27
GSNR w. $\mathbf{L}_{8nn}$	✓	39.88	30.27

Appendix

Table 4: Final PSNR (dB) comparison across graph variants and topologies for demosaicing with

n=64^{2}

and

p=0.5\cdot 3\cdot n

Figure 19: Evaluation of GSNR-PnP on a non-optical imaging example (SAR-like data). Top row: ground-truth image $\mathbf{x}^{\ast}$ , back-projection $\mathbf{H}^{\top}\mathbf{y}$ , baseline PnP reconstruction, and GSNR-PnP with $\mathbf{L}_{4\mathrm{nn}}$ and $\mathbf{L}=\mathbf{I}$ , with PSNR values overlaid. Bottom: PSNR as a function of iteration for GSNR-PnP with $\mathbf{L}_{4\mathrm{nn}}$ , GSNR-PnP with $\mathbf{L}=\mathbf{I}$ , and the PnP baseline.

Figure 20: Super-resolution results with GSNR-PnP for different graph Laplacians and denoisers. Left: PSNR versus iteration for GSNR-PnP with $\mathbf{L}=\mathbf{I}$ , $\mathbf{L}_{4\mathrm{nn}}$ , $\mathbf{L}_{8\mathrm{nn}}$ , and the PnP, using (from left to right) DnCNN, DRUNET, and a wavelet denoiser. Right: corresponding reconstructions for a representative face image, with the ground truth at the top-left and PSNR values overlaid on each result.

Figure 21: Super-resolution on CelebA (20 images) with Deep Image Prior (DIP). We plot PSNR versus iteration for the standard DIP baseline (red) and the proposed GSNR-DIP variants using different Laplacians:

\mathbf{L}=\mathbf{I}

\mathbf{L}_{4\mathrm{nn}}

, and

\mathbf{L}_{8\mathrm{nn}}

Parameter	Lip-DnCNN	Wavelet
$\gamma_{\mathrm{g}}$	0.1	0.1
$\gamma$	0.1	0.1
$\eta$	0.0001	0.001
$\alpha$	$\frac{1.5}{\\|\mathbf{H}\\|}$	$\frac{1.5}{\\|\mathbf{H}\\|}$
$K$ (Max iterations)	800	800

Appendix

Table 5: Experimental settings for GSNR PnP-PGD in deblurring.

Figure 22: Deblurring visual results comparing PGD-PnP, GSNR with $\mathbf{L}=\mathbf{I}$ (NPN), and GSNR with $\mathbf{L}=\mathbf{L}_{8nn}$ across two datasets and two denoisers. Best results for each denoiser are in bold . Here, $p=0.8$ .

Metric / Resolution	$128^{2}$	$256^{2}$	$512^{2}$
Offline EVD computation (s)	102	1315	22310
Offline RAM (GiB)	0.68	10.88	174.09
PnP runtime (Wavelet) (s)	1.09	1.42	1.96

Appendix

Table 6: GSNR’s computational cost for SR: offline and online (PnP, one image, 1000 iters).

	Lip-DnCNN	Wavelet
Baseline (PnP-PGD)	31.23	30.39
NPN [ 17 ]	33.05	32.90
GSNR ( $\mathbf{L}_{8\text{nn}}$ )	33.69	33.65

Appendix

Table 7: Final PSNR on DIV2K for deblurring (

\mathrm{G}

trained on Places365).

Figure 23: PSNR with

\mathrm{G}

ablation in Places for the deblurring task.

Figure 24: PSNR in deblurring with inexact forward operator.

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Laplacian-regularized minimization is fundamental in signal processing and machine learning, but explicit construction of the dense graph Laplacian pseudoinverse can be computationally expensive. To address this issue, we consider the setting where the graph Laplacian is given and propose a Difference-of-Convex Regularizer (DCR) graph learning framework that learns a reusable approximation of the pseudoinverse action through regularized Maximum Likelihood Estimation (MLE) and shrinkage-regularized Convex--Concave Procedure (CCCP) iterations. For Laplacian-Regularized Nonnegative Least Squares (LR-NNLS), dual and Karush-Kuhn-Tucker (KKT) analysis motivates separating graph-dependent pseudoinverse learning from instance-specific optimization. The learned pseudoinverse is embedded into a differentiable primal optimization procedure as a graph-aware preconditioner for nonnegative reconstruction and can be reused across LR-NNLS instances sharing the same graph. We establish stability and fixed-point existence guarantees for the proposed pseudoinverse-learning iteration. Experiments across different graph topologies and scales demonstrate near-reference accuracy, competitive time-to-moderate-accuracy behavior, and effective cross-instance reuse across independently generated problem instances sharing the same graph.

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