Livin' on a Prior: Likelihood Score Approximation for Inverse Problems

Rostislav Makarov; Tal Peer; Danilo de Oliveira; Timo Gerkmann

cs.LGSep 28, 2026

Livin' on a Prior: Likelihood Score Approximation for Inverse Problems

Authors: Rostislav Makarov, Tal Peer, Danilo de Oliveira, Timo Gerkmann

Organizations: Signal Processing University of Hamburg Hamburg, Germany

Abstract

Generative models have found great success as data-driven methods of solving inverse problems. Two popular approaches work either by combining a pretrained generative prior with a known degradation model, or by training a conditional generative model directly from paired data. We target a setting that spans both regimes: unknown degradations can be learned from few paired examples, while known degradations can be learned from self-generated samples. We introduce Likelihood Score Approximation (LSA), a generative framework that keeps a pretrained unconditional model fixed and learns an observation-conditioned model that approximates the likelihood score from paired samples. Within a conditional stochastic-interpolant framework, LSA can be trained in either score or velocity coordinates, independently of the unconditional model's native parameterization, and supports both deterministic and stochastic sampling. We further show empirically that the prior model can be swapped post-training while keeping the same LSA model. Across speech and image inverse problems, LSA operates effectively even at roughly 0.01% of the full training dataset. On the ImageNet-256 benchmark it achieves competitive or better restoration quality than strong posterior-sampling baselines while requiring up to several orders of magnitude fewer network evaluations.

Figures & tables

Figure 1: Prior and posterior vector fields along the Gaussian probability path. (a) Velocity vectors at the same intermediate state $x_{t}$ , differing by $\kappa_{t}\ell_{t}(x,y)$ . (b) Score vectors for $p_{t}(x)$ and $p_{t}(x\mid y)$ , differing by $\ell_{t}(x,y)$ .

Figure 2: Overview of LSA . Training uses either self-generated pairs with a known degradation operator or limited paired-data with an unknown operator. Prior and LSA parameterizations can be chosen independently.

Figure 3: ImageNet- $512$ $8\times$ super-resolution across paired-data budgets, following the CSE benchmark ( Xu et al., 2025 ) . (a) FID for models trained with 25–1k real pairs, together with the self-generated setting. (b) Qualitative results for LSA .

EARS-WHAM-v2
Data	Model	PESQ $\uparrow$	SI-SDR $\uparrow$	Avg. nMOS $\uparrow$	FAD $\downarrow$
1 min	Direct Posterior	1.44	8.70	2.60	1.95
	ControlNet	1.48	9.66	3.33	0.74
	LSA (ours)	1.63	9.80	3.60	0.58
10 min	Direct Posterior	1.70	11.50	3.44	0.25
	ControlNet	1.47	10.18	3.43	0.42

Table 1: Speech restoration results. (E), (U), and (VB) denote EARS, Urgent, and VoiceBank priors. ∗ AnyEnhance is trained on 44.1-kHz audio. Bold : best within each data regime.

Task	Method	NFE	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	FID $\downarrow$	Task	Method	NFE	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	FID $\downarrow$
Super- resolution ( $4\times$ )	DMAP	$\sim$ 300	25.39	.661	.229	74.65	Gaussian deblurring	DAPS	1k	26.15	.684	.253	75.68
	DAPS	1k	25.89	.694	.276	83.57		ControlNet	26	25.08	.668	.242	99.82
	ControlNet	26	25.06	.675	.237	92.15		LSA (ours)	21	25.39	.690	.212	69.34
	LSA (ours)	23	25.57	.700	.215	70.38	Motion deblurring	RePS	1k	28.95	.801	.169	53.15
Inpainting (box)	MAS	–	21.15	.817	.168	95.96		DAPS †	1k	–	.769	.175	47.09
	DAPS	1k	21.43	.725	.214	109.85		LSA (ours)	42	28.18	.802	.130	32.45

Table 2: ImageNet-

256

restoration on 100 images with additive Gaussian noise (

\sigma_{y}=0.05

), following the DAPS benchmark ( Zhang et al., 2025a ) . Bold : best. † DAPS rerun from DAPS++ ( Chen et al., 2026 ) .

Figure 4: ImageNet-

256

restoration with 128 real pairs. All paired methods are trained on the same fixed 128 pairs, with DEFT ( Denker et al., 2024 ) included where available. For reference, we also show LSA trained on self-generated prior samples paired with observations from the known degradation operator.

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Setting	Model	Width	Depth config.	Architecture details	Frozen Parameters	GFLOPs/ NFE
ImageNet- $256$	PixelDiT-XL ( $320$ ep.)	1152	26	4 pixel blocks; patch size 16	797.38M	311.19
ImageNet- $512$	PixelDiT-XL	1152	26	4 pixel blocks; patch size 16	797.38M	1352.24
Speech	NCSN++	128	$(1,1,2,2,2,2,2)$	2 residual blocks	65.56M	132.70
	ADM	128	$(1,1,2,2,2,2,2)$	2 residual blocks	72.25M	131.42

Appendix

Table 3: Prior model configurations used throughout the experiments. All prior parameters remain fixed during downstream training. Speech GFLOPs counts are measured for one second of audio.

Setting	Model	Width	Depth config.	Architecture details	Trainable params	GFLOPs/ NFE
IN256, 0 real pairs	LSA (NCSN++)	40	$(1,1,1,2)$	2 residual blocks	2.16M	44.30
IN256, 128 pairs	LSA (NCSN++)	32	$(1,1,1,2)$	1 residual block	1.01M	19.84
IN256, 128 pairs	ControlNet	692	13	13 control blocks	124.92M	46.80
IN256, 128 pairs	LoRA	–	30	rank=8	2.15M	1.10
IN256 ablation	LSA (PixelDiT)	96	4	2 pixel blocks; patch size 4	0.98M	44.71
IN512, 0 real pairs	LSA (NCSN++)	40	$(1,1,1,2)$	2 residual blocks	2.16M	176.04

Appendix

Table 4: Model configurations used across all ImageNet experiments. Parameters and GFLOPs/NFE refer to the trainable LSA model or adapter.

Setting	Model	Width	Depth config.	Architecture details	Trainable params	GFLOPs/ NFE
1 min	Conditional / LSA	32	$(1,2,2)$	2 residual blocks	1.71M	27.60
10 min	Conditional / LSA	64	$(1,1,2,2)$	1 residual block	5.47M	46.51
Full	Conditional / LSA	128	$(1,1,2,2,2,2,2)$	2 residual blocks	65.59M	266.03
0 real pairs Phase Retrieval	LSA	64	$(1,1,2,2)$	2 residual blocks	7.60M	66.22
1 / 10 min	ControlNet	32	7/7	–	1.81M	5.55
Full	ControlNet	64	7/7	–	6.66M	19.64

Appendix

Table 5: Speech model configurations. Parameters and GFLOPs/NFE refer to the trainable model shown in each row. Speech compute is measured for one second of audio.

Task	Method	NFE	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	FID $\downarrow$
Super-resolution ( $4\times$ )	DEFT	100	24.87	.651	.244	97.15
	ControlNet	43	24.78	.665	.247	89.14
	LSA (ours)	21	24.93	.683	.229	73.01
Inpainting (box)	DEFT	100	19.20	.759	.194	141.19
	ControlNet	10	20.24	.685	.275	152.02
	LSA (ours)	22	18.94	.755	.193	134.00

Appendix

Table 6: Detailed ImageNet-

256

restoration results using 128 real pairs. We report NFE, PSNR, SSIM, LPIPS, and FID across all evaluated tasks. Bold denotes the best result for each task.

Figure 5: ImageNet-256 blur restoration: references, observations, and LSA (ours) reconstructions.

Figure 6: ImageNet-256 super-resolution, inpainting, and HDR restoration results.

Source	Files	NFE $\downarrow$	PSNR $\uparrow$	LPIPS $\downarrow$	FID $\downarrow$
DPS+CSE	25	500	22.26	.405	75.64
LSA (ours)	25	12	23.07	.347	40.76
DPS+CSE	50	500	22.84	.353	55.78
LSA (ours)	50	9	23.20	.335	39.34
CSE	100	500	16.54	.484	85.58
DPS+CSE	100	500	22.98	.323	44.59

Appendix

Table 7: ImageNet-

512

8\times

super-resolution across paired-data budgets. LSA is trained with 25, 50, 100, or 1k real pairs; “self-gen” denotes training on self-generated pairs obtained using the known degradation operator. Baseline results are from Xu et al. (2025) . Bold indicates the best available value within each data budget.

Parameterization	Initialization	$t_{\mathrm{start}}$	NFE	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	FID $\downarrow$
Score	Observation marginal	.878	41	24.91	.663	.217	72.86
Velocity	Gaussian	.907	49	24.68	.657	.216	70.23
Velocity	Gaussian	1.000	36	24.86	.661	.221	73.15
Data prediction	Observation marginal	.819	31	25.10	.676	.229	75.72
Data prediction	Gaussian	1.000	29	24.49	.655	.233	74.19

Appendix

Table 8: LSA parameterization ablation for Gaussian deblurring on ImageNet-

256

with 128 pairs and an NCSN++ backbone. For each parameterization, we report the validation-selected sampling start time

t_{\mathrm{start}}

and initialization. For velocity and data prediction, we additionally report default sampling from the Gaussian endpoint at

t_{\mathrm{start}}=1

; this setting is not defined for the score parameterization due to the endpoint singularity.

FM backbone	Initialization	$t_{\mathrm{start}}$	NFE	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	FID $\downarrow$
NCSN++	Gaussian	.907	49	24.68	.657	.216	70.23
NCSN++	Gaussian	1.000	36	24.86	.661	.221	73.15
PixelDiT	Gaussian	.840	22	25.25	.682	.213	68.31
PixelDiT	Gaussian	1.000	19	24.97	.673	.217	71.52

Appendix

Table 9: LSA backbone ablation for Gaussian deblurring on ImageNet-

256

with 128 pairs. We compare NCSN++ and PixelDiT using Gaussian initialization, reporting both the validation-selected sampling start time

t_{\mathrm{start}}

and sampling from the Gaussian endpoint at

t_{\mathrm{start}}=1

. (Appendix B.4 )

Figure 7: Gaussian-deblurring learning curves for LSA on ImageNet- $256$ with a PixelDiT prior, trained using either 128 real pairs or self-generated data. Training time is measured on a single A100 GPU.

Figure 8: Qualitative BWE comparison on three EARS test utterances. The shared top rows show the clean reference and low-pass-filtered observation; each subsequent block compares ControlNet, direct posterior training, and LSA (ours) using 1 minute, 10 minutes, or the full paired training set. Spectrogram magnitudes are measured relative to the clean-reference peak of each utterance, with frequency shown in kHz.

Figure 9: Inference-time prior swapping without retraining LSA . Top: ImageNet- $256$ Gaussian deblurring with 128 pairs, where the PixelDiT prior used during training (epoch 320) is replaced by earlier checkpoints. Bottom: 1-min speech-enhancement LSA , where the NCSN++ training prior is replaced by an ADM prior trained on the same EARS data. Values show metric changes relative to the training prior; despite the architecture change, performance remains broadly comparable, with a substantial improvement in LSD.

Figure 10: ODE versus SDE sampling for LSA bandwidth extension with 1 min, 10 min, and full paired data. Changes are relative to the ODE baseline ( $\rho=0$ ); SDE uses $\rho=0.18$ , $0.09$ , and $0.13$ , respectively. Mild stochasticity generally improves performance, providing complementary gains.

Restart	$\zeta_{1}{\rightarrow}\zeta_{0}$	PESQ $\uparrow$	SI-SDR $\uparrow$	Avg. nMOS $\uparrow$	FAD $\downarrow$
Yes	$3{\rightarrow}0$	1.51	7.82	3.81	0.35
No	$3{\rightarrow}0$	1.52	7.67	3.47	1.29
Yes	$1$	1.47	7.43	3.14	0.70
Yes	$2$	1.51	10.56	2.93	1.82
No	$1$	1.36	5.96	2.76	1.11

Appendix

Table 10: Inference-refinement ablation for the 1-min LSA speech-enhancement model. We vary the use of restart refinement and the LSA guidance-weight schedule

\zeta_{1}{\rightarrow}\zeta_{0}

, while keeping the trained model fixed (see Appendix B.4 ).

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Exact Posterior Score Estimation for Solving Linear Inverse Problems

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Bayesian Inverse Problems Posterior Sampling Iterative Denoising Methods Restoration Scores

Open paper AskPDF arXiv

Diffusion and flow-based models learn powerful data priors by training a denoiser to reverse Gaussian corruption. To use this prior to solve a linear inverse problem, one needs to sample from the posterior, but the score that the prior provides is the unconditional score, not the posterior score. Existing methods either steer a fixed pretrained denoiser with approximate measurement-matching corrections, or train a conditional restoration model that abandons the denoising structure of the prior. We derive the exact posterior score in closed form for linear Gaussian inverse problems under general Gaussian interpolants, and show that posterior sampling reduces to a denoising problem at an operator-dependent shifted pivot under an anisotropic noise covariance. We turn this identity into Exact Posterior Score (EPS), a denoising training objective that preserves the input/output structure of standard pretraining and can therefore be trained from scratch or fine-tuned from a pretrained denoiser. At inference, EPS uses the same sampler as the underlying backbone, with no likelihood gradients or projections. We evaluate EPS on five linear inverse problems across FFHQ and ImageNet, where it outperforms training-free and training-based baselines on fidelity, perceptual, and distributional metrics, while using roughly an order of magnitude fewer denoiser evaluations than gradient-based posterior samplers.

Abstract

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