FLINT: Coupling Proximal Initialization and Bounded Stochasticity for Flow-Matching Inverse Problems

Junseo Bang; Daewon Choi; Dong Ju Mun; Se Young Chun

cs.CVNov 4, 2026

FLINT: Coupling Proximal Initialization and Bounded Stochasticity for Flow-Matching Inverse Problems

Authors: Junseo Bang, Daewon Choi, Dong Ju Mun, Se Young Chun

Organizations: Dept. of ECE, Seoul National University, Republic of Korea · IPAI, Seoul National University, Republic of Korea · INMC & AIIS, Seoul National University, Republic of Korea

Abstract

Flow matching models have emerged as a dominant paradigm for generative posterior sampling in imaging inverse problems. Existing solvers follow two distinct lines of work: trajectory-space methods, which refine sampling dynamics from random initial noise, and source-space methods, which optimize the initial noise itself. While most prior work treats these as independent design choices, we show that they are inherently coupled. We derive a unified stability bound that links initialization proximity and cumulative sampling stochasticity through a multiplicative envelope, complemented by two empirical observations. First, we identify a sufficiency regime in which a moderately optimized proximal source initialization is enough to attain the performance plateau, rendering exhaustive source-space optimization unnecessary. Second, this initial advantage is fragile: it is eroded by excessive stochasticity accumulation during guided sampling, necessitating a mechanism that bounds the cumulative noise injection. These findings motivate FLINT, a two-stage framework that operationalizes this coupling. FLINT establishes a high-fidelity starting point via a manifold-constrained proximal objective in only a few iterations, then performs guided sampling under a bounded stochasticity schedule that helps prevent erasure of the initialization advantage. Across diverse inverse problems, FLINT delivers state of the art perceptual quality and competitive distortion metrics while maintaining superior computational efficiency.

Figures & tables

Figure 1: Illustration of generative paradigms for inverse problems . (a) and (b) depict trajectory-space methods, while panels (c) and (d) depict source-space optimization methods. (a) Likelihood score corrections are applied directly at each noisy intermediate state $\mathbf{x}_{t}$ . (b) Likelihood score corrections are applied in clean image space via the Tweedie estimate $\hat{\mathbf{x}}_{0|t}$ , followed by renoising to the generative manifold at the next timestep. (c) Source-space MAP methods reparameterize the clean image via the deterministic generative map $\Phi$ as $\mathbf{x}_{0}=\Phi(\mathbf{x}_{1})$ and optimize over the initial noise $\mathbf{x}_{1}$ to minimize data fidelity. (d) Source-space posterior sampling draws samples from $p(\mathbf{x}_{1}\mid\mathbf{y})$ via Hamiltonian Monte Carlo or Langevin dynamics. (e) By strategically coupling proximal source initialization with bounded stochasticity during guided posterior sampling, FLINT bounds the sampling trajectory, preserving the initialization advantage and yielding high-fidelity convergence.

Figure 2: Interplay of proximal initialization and sampling stochasticity on Gaussian deblurring. (a) PSNR and LPIPS as functions of the perturbation scale $C_{0}$ across varying the cumulative stochasticity $G$ , demonstrating the existence of a sufficiency regime ( $C_{0}\leq 0.4$ in this setup) where proximal initialization performs on par with exact inversion. (b) Visual comparisons for $C_{0}\in\{0,0.4,\pi/2\}$ . While a representative proximal initialization within this regime ( $C_{0}=0.4$ ) matches the high fidelity of exact inversion under posterior sampling ( $G=0.008$ ), it preserves measurement-consistent structures even in unconditional sampling, unlike the unrelated output from random initialization.

	Super-Resolution $\times$ 8				Super-Resolution $\times$ 12				Motion Deblurring				Gaussian Deblurring
Method	PSNR $\uparrow$	SSIM $\uparrow$	FID $\downarrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	FID $\downarrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	FID $\downarrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	FID $\downarrow$	LPIPS $\downarrow$
FFHQ (768 $\times$ 768)
FlowChef	27.94	0.773	104.19	0.438	26.21	0.725	170.39	0.554	25.42	0.733	88.78	0.478	26.62	0.748	103.13	0.488
FlowDPS	27.08	0.711	28.91	0.379	26.66	0.736	30.55	0.354	25.02	0.688	45.54	0.427	25.80	0.699	47.21	0.420
FLAIR	27.58	0.697	29.73	0.285	26.77	0.709	24.75	0.307	29.39	0.755	13.83	0.259	28.68	0.744	15.24	0.259
Flower	28.36	0.793	35.97	0.351	27.15	0.762	76.44	0.444	26.38	0.756	71.93	0.4410	26.96	0.759	86.03	0.462

Table 1: Quantitative comparison on a flow matching model averaged over 1,000 FFHQ and 800 DIV2K samples using 50 NFEs under Gaussian noise (

\sigma_{y}=0.03

Figure 3: Qualitative comparison for FFHQ and DIV2K datasets.

Figure 4: Computational cost and performance. PSNR and LPIPS vs. sampling time (sec) for SR $\times$ 8 and Gaussian deblurring. Numbers after method names denote its NFEs.

	Super-Resolution $\times$ 4				Motion Deblurring
Method	PSNR $\uparrow$	SSIM $\uparrow$	FID $\downarrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	FID $\downarrow$	LPIPS $\downarrow$
PSLD	26.07	0.712	46.69	0.309	25.06	0.703	59.61	0.367
ReSample	25.96	0.849	50.97	0.225	24.18	0.765	92.86	0.397
DDPG	29.73	0.809	84.12	0.305	26.62	0.752	94.41	0.358
TReg	25.22	0.756	37.35	0.244	25.42	0.701	45.38	0.274
Ours	29.88	0.776	22.36	0.161	30.05	0.791	33.32	0.182

Table 2: Quantitative comparison on a diffusion model over 1,000 FFHQ samples using 200 NFEs under additive Gaussian noise (

\sigma_{y}=0.03

Table 3: Ablation studies on Gaussian deblurring (FFHQ, 100 samples). (a) Stage 1: Components include MCPO and SWDC. (b) Stage 2: Performance of the BSS profile across

\lambda_{s}

Figure 5: Initialization quality on SR $\times$ 8. Unconditional sampling from the Stage 1 output (third column) demonstrates that FLINT’s optimized noise recovers faithful coarse structures, contrasting with the unrelated results from random initialization (second column). Starting from this informed point, Stage 2 achieves high-fidelity reconstruction. Metrics are averaged over 100 FFHQ samples.

Figure 6: Quantitative comparison of the sampling process via Tweedie’s estimates.

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Figure S1 : Additional empirical validation for erosion by excessive stochasticity. (a) PSNR and LPIPS as a function of the cumulative stochasticity $G$ for proximal initialization ( $C_{0}=0.4$ ) and random initialization ( $C_{0}=\pi/2$ ). Proximal initialization outperforms random initialization at low $G$ , while the performance gap narrows as $G$ increases. (b) Qualitative comparison across initialization types (proximal vs. random, rows) and stochasticity levels (high $G=0.12$ vs. low $G=0.008$ , columns).

Figure S2 : Qualitative comparison for FFHQ dataset on super-resolution 4 $\times$ and motion deblurring based on diffusion model (SD 1.5).

Framework	Dataset	$\beta_{\mathrm{KL}}$	$\lambda_{\mathrm{cum}}$	$\sigma_{\max}$	$\alpha_{\mathrm{lr}}$
Flow Matching (SD3.5-M)	FFHQ	0.1	10.0	0.5	0.006
Flow Matching (SD3.5-M)	DIV2K	0.01	1.0	0.5	0.01
Diffusion (SD1.5)	FFHQ	0.01	10.0	0.02	0.02

Appendix

Table S1 : Stage 1 hyperparameters across datasets and generative frameworks.

Framework	Dataset	SR $\times$ 8	SR $\times$ 12	Motion Deblurring	Gaussian Deblurring
Flow Matching (SD3.5-M)	FFHQ	0.9	0.9	0.6	0.8
Flow Matching (SD3.5-M)	DIV2K	0.9	0.8	0.9	0.9
Framework	Dataset	SR $\times$ 4	Motion Deblur
Diffusion (SD1.5)	FFHQ	0.3	0.3

Appendix

Table S2 : Stochasticity magnitude

\lambda_{s}

across tasks and datasets.

Framework	Dataset	SR $\times$ 8	SR $\times$ 12	Motion Deblurring	Gaussian Deblurring
Flow Matching (SD3.5-M)	FFHQ	(70, 130)	(30, 130)	(110, 130)	(60, 140)
Flow Matching (SD3.5-M)	DIV2K	(125, 50)	(120, 0)	(450, 250)	(160, 100)
Framework	Dataset	SR $\times$ 4	Motion Deblur
Diffusion (SD1.5)	FFHQ	(10, 100)	(50, 20)

Appendix

Table S3 : Data consistency guidance scale

(\beta_{\min},\beta_{\max})

configurations across tasks and datasets.

	FFHQ (768 $\times$ 768)				DIV2K (768 $\times$ 768)
Method	PSNR $\uparrow$	SSIM $\uparrow$	FID $\downarrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	FID $\downarrow$	LPIPS $\downarrow$
FlowChef	22.82	0.785	57.23	0.375	20.01	0.606	70.08	0.463
FlowDPS	21.74	0.783	33.56	0.345	20.36	0.618	29.97	0.415
FLAIR	28.34	0.867	11.86	0.141	23.74	0.826	12.09	0.167
Flower	27.12	0.811	37.94	0.334	22.66	0.687	36.36	0.359
FlowLPS	26.65	0.829	9.59	0.249	22.28	0.739	8.17	0.257

Appendix

Table S4 : Quantitative comparison on inpainting based on flow matching model. All metrics are averaged over 1,000 FFHQ and 800 DIV2K samples, with each method allocated 50 NFEs under additive Gaussian noise (

\sigma_{y}=0.03

Figure S3 : Qualitative comparison for DIV2K datasets on inpainting task.

Flow Matching Model (SD3.5-M)
Method	SR $\times$ 8 Runtime(s) $\downarrow$	SR $\times$ 12 Runtime(s) $\downarrow$	Motion Deblur Runtime(s) $\downarrow$	Gaussian Deblur Runtime(s) $\downarrow$
FlowChef	8.14	8.14	9.28	8.66
FlowDPS	8.22	8.18	9.47	8.71
FLAIR	20.33	20.28	22.94	21.37
Flower	16.89	16.83	23.36	19.46
FlowLPS	16.84	16.78	22.82	19.31

Appendix

Table S5 : Runtime comparison of methods based on flow matching and diffusion models. Sampling time per image, in seconds, is averaged over 10 samples.

Figure S4 : Qualitative comparison of the Tweedie estimate’s along the sampling trajectory for FLINT and competing baselines.

Method	PSNR $\uparrow$	SSIM $\uparrow$	KID $\downarrow$	LPIPS $\downarrow$
$\lambda_{\mathrm{cum}}=0$ (OFF)	30.32	0.802	0.015	0.263
$\lambda_{\mathrm{cum}}=10.0$ (ON)	30.39	0.800	0.008	0.224

Appendix

Table S6 : Ablation on cumulant regularizer

\mathcal{R}(\mu_{\phi})

coefficient

\lambda_{\mathrm{cum}}

for Motion Deblurring, averaged over 100 FFHQ samples under Gaussian noise (

\sigma_{y}=0.03

	Gaussian Deblurring
Method	PSNR $\uparrow$	SSIM $\uparrow$	KID $\downarrow$	LPIPS $\downarrow$
Forward-Noising	28.64	0.616	0.020	0.449
Deterministic-Inv.	29.97	0.788	0.021	0.231
Stochastic-Inv.	29.81	0.751	0.010	0.209
Ours	30.11	0.788	0.005	0.192

Appendix

Table S7 : Quantitative comparison of alternative initialization strategies that derive the initial noise from the pseudo-inverse measurement, with Stage 2 fixed. Gaussian deblurring (FFHQ, 100 samples).

Figure S5 : Effect of excessive source-space optimization on Gaussian deblurring (FFHQ, 100 samples). (a) PSNR and LPIPS of FLINT reconstructions across noise optimization steps. Beyond the proximal regime (10 steps), PSNR plateaus while LPIPS degrades by over $35\%$ , as the OOD noise can no longer be rendered into natural image structures by the pretrained generative model. (b) PSNR and LPIPS of unconditional sampling rendered directly from the optimized noise (NFE = 29, matched to Stage 2). (c) PSNR, LPIPS, and horizontal lag-1 spatial correlation of the optimized noise on a representative sample. The correlation remains stable within the proximal regime and trends upward beyond it, indicating a gradual drift toward the boundary of the prior’s typical set.

Figure S6 : Posterior variance analysis for super-resolution $\times$ 12. For each method, 50 samples are drawn from the posterior $p(\mathbf{x}|\mathbf{y})$ for a fixed observation $\mathbf{y}$ , with four representative samples, the posterior mean $\mathbb{E}[\mathbf{x}|\mathbf{y}]$ , and the pixel-wise standard deviation $\text{Std}[\mathbf{x}|\mathbf{y}]$ displayed. FLINT produces diverse posterior samples with meaningful variance in perceptually significant regions, confirming the absence of mode collapse.

Figure S7 : Additional qualitative comparison for the super resolution tasks (super-resolution $\times$ 8 and super-resolution $\times$ 12) on the FFHQ dataset.

Figure S8 : Additional qualitative comparison for the deblurring tasks (motion deblurring and gaussian deblurring) on the FFHQ dataset.

Figure S9 : Additional qualitative comparison for the super resolution tasks (super-resolution $\times$ 8 and super-resolution $\times$ 12) on the DIV2K dataset.

Figure S10 : Additional qualitative comparison for the deblurring tasks (motion deblurring and gaussian deblurring) on the DIV2K dataset.

Figure S11 : Qualitative failure cases of FLINT on highly complex scenes (DIV2K dataset). Increased ill-posedness in dense, multi-object images disrupts the balance between measurement fidelity and the generative prior. This results in either hallucinated structures, where the model generates spurious details to satisfy the prior, or blurry textures, where conflicting updates fail to resolve high-frequency information. These cases highlight the inherited limitations of the pretrained backbone’s capacity for complex out-of-distribution scenes.

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Flow matching has emerged as the state-of-the-art generative model and has been used for plug-and-play (PnP) priors to solve inverse problems in computational imaging. However, existing flow-based inverse solvers make simplifying approximations in posterior sampling and/or assume linear forward models. In contrast, principled diffusion-based methods exist but are computationally expensive. To circumvent these problems, we introduce FlowSGS, a flow-based posterior sampling method using Split Gibbs Sampling (SGS) with annealed coupling, and make flow-specific interventions to the prior sampling and annealing steps, theoretically grounded in the Stochastic Interpolants (SI) framework. Using a novel timestep correction and accelerated warm-up with one-step flow models, our method is faster and more accurate than the state of the art across a range of imaging problems. We demonstrate the first use of flow-based samplers for a nonlinear inverse problem (Fourier phase retrieval) and apply one-step flow models to principled Bayesian inverse imaging.

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