Wenxuan Yuan

Xiaoyuan Cheng, Wenxuan Yuan, Yiming Yang et al.

The Koopman operator provides a powerful framework for modeling dynamical systems and has attracted growing interest from the machine learning community. However, its infinite-dimensional nature makes identifying suitable finite-dimensional subspaces challenging, especially for deep architectures. We argue that these difficulties come from suboptimal representation learning, where latent variables fail to balance expressivity and simplicity. This tension is closely related to the information bottleneck (IB) dilemma: constructing compressed representations that are both compact and predictive. Rethinking Koopman learning through this lens, we demonstrate that latent mutual information promotes simplicity, yet an overemphasis on simplicity may cause latent space to collapse onto a few dominant modes. In contrast, expressiveness is sustained by the von Neumann entropy, which prevents such collapse and encourages mode diversity. This insight leads us to propose an information-theoretic Lagrangian formulation that explicitly balances this tradeoff. Furthermore, we propose a new algorithm based on the Lagrangian formulation that encourages both simplicity and expressiveness, leading to a stable and interpretable Koopman representation. Beyond quantitative evaluations, we further visualize the learned manifolds under our representations, observing empirical results consistent with our theoretical predictions. Finally, we validate our approach across a diverse range of dynamical systems, demonstrating improved performance over existing Koopman learning methods. The implementation is publicly available at https://github.com/Wenxuan52/InformationKoopman.

Yiming Yang, Xiaoyuan Cheng, Yi He et al.

Diffusion models have recently emerged as powerful learned priors for Bayesian inverse problems (BIPs). Diffusion-based solvers rely on a presumed likelihood for the observations in BIPs to guide the generation process. However, the link between likelihood and recovery quality for BIPs is unclear in previous works. We bridge this gap by characterizing the posterior approximation error and proving the \emph{stability} of the diffusion-based solvers. Meanwhile, an immediate result of our findings on stability demonstrates the lack of robustness in diffusion-based solvers, which remains unexplored. This can degrade performance when the presumed likelihood mismatches the unknown true data generation processes. To address this issue, we propose a simple yet effective solution, \emph{robust diffusion posterior sampling}, which is provably \emph{robust} and compatible with existing gradient-based posterior samplers. Empirical results on scientific inverse problems and natural image tasks validate the effectiveness and robustness of our method, showing consistent performance improvements under challenging likelihood misspecifications.

Xiaoyuan Cheng, Wenxuan Yuan, Boyang Li et al.

Diffusion policy sampling enables reinforcement learning (RL) to represent multimodal action distributions beyond suboptimal unimodal Gaussian policies. However, existing diffusion-based RL methods primarily focus on offline settings for reward maximization, with limited consideration of safety in online settings. To address this gap, we propose Augmented Lagrangian-Guided Diffusion (ALGD), a novel algorithm for off-policy safe RL. By revisiting optimization theory and energy-based model, we show that the instability of primal-dual methods arises from the non-convex Lagrangian landscape. In diffusion-based safe RL, the Lagrangian can be interpreted as an energy function guiding the denoising dynamics. Counterintuitively, direct usage destabilizes both policy generation and training. ALGD resolves this issue by introducing an augmented Lagrangian that locally convexifies the energy landscape, yielding a stabilized policy generation and training process without altering the distribution of the optimal policy. Theoretical analysis and extensive experiments demonstrate that ALGD is both theoretically grounded and empirically effective, achieving strong and stable performance across diverse environments.