control-guided sampling

This study addresses the computational bottleneck associated with dynamically constrained sampling of state spaces in feedback control and planning. To overcome this challenge, the work reformulates control as a dynamically constrained sampling problem, establishing a mapping framework that bridges controllability, optimal control theory, and generative modeling. Specifically, it integrates flow matching, normalizing flows, and denoising diffusion techniques to guide system evolution toward target states or distributions. The proposed approach enables efficient reachable set sampling and precise trajectory planning while unifying control-theoretic and generative-modeling paradigms. Furthermore, the authors provide an accessible open-source tutorial to facilitate practical adoption by the research community.

Diffusion models suffer from weak controllability during sampling, making it difficult to satisfy statistical constraints—primarily because the relationship between initial noise perturbations and final outputs remains uncharacterized. This work provides the first theoretical proof that, under diffusion ODE sampling, the output exhibits a strongly linear response to initial noise perturbations. Leveraging this insight, we propose CCS (Controlled Controllable Sampling), the first explicit noise-space controllable sampling framework. CCS employs differentiable controllers in the noise space to precisely specify target statistics—e.g., mean and variance—without altering the model architecture or requiring retraining. Evaluated across multiple benchmarks, CCS achieves state-of-the-art statistical controllability while preserving high sample quality (FID ≤ 2.1) and diversity (LPIPS ≥ 0.43).

This work addresses the challenge of balancing sampling efficiency and reconstruction quality in diffusion-based image inverse problems by modeling the denoising process as a dynamical system. The authors propose a closed-form, stepwise guidance mechanism grounded in stochastic optimal control, which explicitly injects a control signal at each sampling step to steer the clean prediction toward consistency with observed measurements. Unlike trajectory-level optimization approaches, this method operates incrementally, substantially reducing computational overhead. By adaptively tuning the control strength to align with the diffusion model’s prior capabilities, the approach achieves high measurement consistency while enhancing perceptual quality. Extensive experiments demonstrate that the proposed method consistently attains superior visual fidelity and quantitative performance across a range of image inverse tasks.

Existing diffusion-based posterior sampling methods struggle to ensure sampling accuracy and reliable uncertainty quantification under nonlinear operators or multimodal posteriors due to heuristic guidance approximations. This work proposes a path-space stochastic optimal control framework that formulates posterior sampling as learning a controlled stochastic process whose trajectory distribution matches the likelihood-weighted target measure. By employing a time reparameterization, the method eliminates bias arising from the unknown initial value function without requiring auxiliary training. It unifies guidance-based sampling and control learning, and incorporates importance sampling correction to yield asymptotically unbiased estimates of posterior expectations. Evaluated on multiple inverse problem benchmarks, the proposed approach significantly outperforms existing methods in sampling accuracy, robustness, and uncertainty quantification.

This work addresses the challenges of sampling and fine-tuning diffusion and flow models under exponentially tilted target distributions by proposing a unified framework grounded in stochastic optimal control and nonequilibrium thermodynamics. The framework integrates adjoint matching with a novel score-matching approach, introduces a bias-variance decomposition to characterize gradient variance properties, and establishes norm bounds for the adjoint ordinary differential equation. It further extends CMCD/NETS losses and the Crooks/Jarzynski identities to the exponentially tilted setting. Experiments on Stable Diffusion 1.5 and 3 demonstrate the efficacy of the proposed method, showing that adjoint-based approaches not only exhibit finite gradient variance but also significantly outperform baseline methods in reward fine-tuning and unnormalized density sampling tasks.

Diffusion posterior samplers are widely used in inverse problems, yet their outputs suffer from bias and discretization instability at low temperatures, with the underlying mechanisms poorly understood. This work constructs a tractable surrogate path bridging the true posterior and a standard Gaussian distribution, leveraging the Feynman–Kac formula to express the density ratio as a path-space expectation. For the first time, it derives a partial differential equation that characterizes sampling bias. By integrating Radon–Nikodym derivatives, Ornstein–Uhlenbeck processes, and auxiliary drift reconstruction, the study reveals the origins and spatial distribution of bias in methods such as DPS and STSL: it precisely identifies regions of over- and under-sampling in DPS and explains how STSL enhances stability through a smoothed reaction term. This theoretical framework provides a foundation for designing stable and efficient posterior sampling algorithms.

This study addresses the weight degeneracy and particle collapse problems in sequential Monte Carlo (SMC) sampling by proposing an interacting particle guidance method that replaces conventional reweighting strategies with a transport mechanism to achieve unbiased sampling. The core innovation lies in introducing a Feynman–Kac drift term that precisely cancels the reweighting term, thereby establishing a fully weight-free sampling framework. This approach deeply integrates diffusion models, closed-form solutions in reproducing kernel Hilbert spaces (RKHS), and interacting particle system theory. Experimental results demonstrate that the proposed method efficiently generates high-quality samples in image inpainting and protein structure inference tasks, significantly enhancing both the practicality and sampling efficiency of SMC.

This work addresses the challenge of achieving efficient, stable, and high-fidelity controllable generation with diffusion models without requiring retraining or backpropagation. By uncovering the intrinsic geometric structure underlying information preservation during the diffusion process, the authors construct a spectral basis derived from the singular functions of the conditional expectation operator. This enables projection of arbitrary guidance signals—such as class labels, CLIP embeddings, or spatial masks—onto the sampling trajectory, yielding training-free, precise control. The method further identifies, for the first time, a phase transition phenomenon within the diffusion process and locates an optimal guidance window, thereby unifying support for multimodal control. On CIFAR-10, it improves conditional accuracy by 37 percentage points over the strongest training-free baseline while accelerating sampling by a factor of four.

This study addresses the reliance of generative model sampling on gradient computation, which limits applicability to black-box or time-varying target distributions. To overcome this limitation, this work proposes a gradient-free sampling framework based on high-frequency cosine perturbations. By extending bounded extremum seeking to Itô processes as a substitute for the drift term, the method enables both MCMC and diffusion sampling for energy-based and score-based models. Notably, it requires no ellipticity assumptions and accommodates non-smooth energy functions. Experiments on CelebA-HQ and CIFAR-10 validate the effectiveness of the approach in latent-space tracking of time-varying images and energy-based model sampling. Overall, this research establishes a new paradigm for generative model sampling in black-box scenarios where gradient information is unavailable or unreliable.

This study addresses the degradation in generation quality and diversity caused by high-strength classifier-free guidance (CFG) in diffusion models, which violates sampling dynamics principles. To resolve this, the work reformulates CFG as a continuous-time optimal control problem, establishing for the first time its theoretical connection to the Hamilton-Jacobi-Bellman (HJB) equation. Building upon this formulation, the authors propose curvature-aware CFG, which introduces hyperspherical control constraints based on Gaussian regularization to correct unconstrained biases and optimize generative trajectories. By aligning the guidance mechanism with principled optimal control theory, this approach significantly enhances image generation quality under moderate-to-high guidance strengths. Ultimately, the proposed method achieves a superior trade-off between sample fidelity and diversity compared to standard CFG, offering a theoretically grounded solution for controllable diffusion-based synthesis.