Jian Xu

Jian Xu, Shian Du, Junmei Yang et al.

Recently, Gaussian processes have been used to model the vector field of continuous dynamical systems, referred to as GPODEs, which are characterized by a probabilistic ODE equation. Bayesian inference for these models has been extensively studied and applied in tasks such as time series prediction. However, the use of standard GPs with basic kernels like squared exponential kernels has been common in GPODE research, limiting the model's ability to represent complex scenarios. To address this limitation, we introduce normalizing flows to reparameterize the ODE vector field, resulting in a data-driven prior distribution, thereby increasing flexibility and expressive power. We develop a data-driven variational learning algorithm that utilizes analytically tractable probability density functions of normalizing flows, enabling simultaneous learning and inference of unknown continuous dynamics. Additionally, we also apply normalizing flows to the posterior inference of GP ODEs to resolve the issue of strong mean-field assumptions in posterior inference. By applying normalizing flows in both these ways, our model improves accuracy and uncertainty estimates for Bayesian Gaussian Process ODEs. We validate the effectiveness of our approach on simulated dynamical systems and real-world human motion data, including time series prediction and missing data recovery tasks. Experimental results show that our proposed method effectively captures model uncertainty while improving accuracy.

Jian Xu, Zhiqi Lin, Min Chen et al.

Deep Gaussian process models typically employ discrete hierarchies, but recent advancements in differential Gaussian processes (DiffGPs) have extended these models to infinite depths. However, existing DiffGP approaches often overlook the uncertainty in kernel hyperparameters by treating them as fixed and time-invariant, which degrades the model's predictive performance and neglects the posterior distribution. In this work, we introduce a fully Bayesian framework that models kernel hyperparameters as random variables and utilizes coupled stochastic differential equations (SDEs) to jointly learn their posterior distributions alongside those of inducing points. By incorporating the estimation uncertainty of hyperparameters, our method significantly enhances model flexibility and adaptability to complex dynamic systems. Furthermore, we employ a black-box adaptive SDE solver with a neural network to achieve realistic, time varying posterior approximations, thereby improving the expressiveness of the variational posterior. Comprehensive experimental evaluations demonstrate that our approach outperforms traditional methods in terms of flexibility, accuracy, and other key performance metrics. This work not only provides a robust Bayesian extension to DiffGP models but also validates its effectiveness in handling intricate dynamic behaviors, thereby advancing the applicability of Gaussian process models in diverse real-world scenarios.

Jian Xu, Zhiqi Lin, Shigui Li et al.

Bayesian Last Layer (BLL) models focus solely on uncertainty in the output layer of neural networks, demonstrating comparable performance to more complex Bayesian models. However, the use of Gaussian priors for last layer weights in Bayesian Last Layer (BLL) models limits their expressive capacity when faced with non-Gaussian, outlier-rich, or high-dimensional datasets. To address this shortfall, we introduce a novel approach that combines diffusion techniques and implicit priors for variational learning of Bayesian last layer weights. This method leverages implicit distributions for modeling weight priors in BLL, coupled with diffusion samplers for approximating true posterior predictions, thereby establishing a comprehensive Bayesian prior and posterior estimation strategy. By delivering an explicit and computationally efficient variational lower bound, our method aims to augment the expressive abilities of BLL models, enhancing model accuracy, calibration, and out-of-distribution detection proficiency. Through detailed exploration and experimental validation, We showcase the method's potential for improving predictive accuracy and uncertainty quantification while ensuring computational efficiency.

Jian Xu, Wei Chen, Shigui Li et al.

Diffusion models have achieved remarkable success in low-light image enhancement through Retinex-based decomposition, yet their requirement for hundreds of iterative sampling steps severely limits practical deployment. While recent consistency models offer promising one-step generation for \textit{unconditional synthesis}, their application to \textit{conditional enhancement} remains unexplored. We present \textbf{Consist-Retinex}, the first framework adapting consistency modeling to Retinex-based low-light enhancement. Our key insight is that conditional enhancement requires fundamentally different training dynamics than unconditional generation standard consistency training focuses on low-noise regions near the data manifold, while conditional mapping critically depends on large-noise regimes that bridge degraded inputs to enhanced outputs. We introduce two core innovations: (1) a \textbf{dual-objective consistency loss} combining temporal consistency with ground-truth alignment under randomized time sampling, providing full-spectrum supervision for stable convergence; and (2) an \textbf{adaptive noise-emphasized sampling strategy} that prioritizes training on large-noise regions essential for one-step conditional generation. On VE-LOL-L, Consist-Retinex achieves \textbf{state-of-the-art performance with single-step sampling} (\textbf{PSNR: 25.51 vs. 23.41, FID: 44.73 vs. 49.59} compared to Diff-Retinex++), while requiring only \textbf{1/8 of the training budget} relative to the 1000-step Diff-Retinex baseline.

Jian Xu, Qibin Zhao, John Paisley et al.

Deep Gaussian processes (DGPs) enable expressive hierarchical Bayesian modeling but pose substantial challenges for posterior inference, especially over inducing variables. Denoising diffusion variational inference (DDVI) addresses this by modeling the posterior as a time-reversed diffusion from a simple Gaussian prior. However, DDVI's fixed unconditional starting distribution remains far from the complex true posterior, resulting in inefficient inference trajectories and slow convergence. In this work, we propose Diffusion Bridge Variational Inference (DBVI), a principled extension of DDVI that initiates the reverse diffusion from a learnable, data-dependent initial distribution. This initialization is parameterized via an amortized neural network and progressively adapted using gradients from the ELBO objective, reducing the posterior gap and improving sample efficiency. To enable scalable amortization, we design the network to operate on the inducing inputs, which serve as structured, low-dimensional summaries of the dataset and naturally align with the inducing variables' shape. DBVI retains the mathematical elegance of DDVI, including Girsanov-based ELBOs and reverse-time SDEs,while reinterpreting the prior via a Doob-bridged diffusion process. We derive a tractable training objective under this formulation and implement DBVI for scalable inference in large-scale DGPs. Across regression, classification, and image reconstruction tasks, DBVI consistently outperforms DDVI and other variational baselines in predictive accuracy, convergence speed, and posterior quality.

Wei Chen, Shigui Li, Jiacheng Li et al.

Estimating density ratios is a fundamental problem in machine learning, but existing methods often trade off accuracy for efficiency. We propose \textit{Interval-annealed Secant Alignment Density Ratio Estimation (ISA-DRE)}, a framework that enables accurate, any-step estimation without numerical integration. Instead of modeling infinitesimal tangents as in prior methods, ISA-DRE learns a global secant function, defined as the expectation of all tangents over an interval, with provably lower variance, making it more suitable for neural approximation. This is made possible by the \emph{Secant Alignment Identity}, a self-consistency condition that formally connects the secant with its underlying tangent representations. To mitigate instability during early training, we introduce \emph{Contraction Interval Annealing}, a curriculum strategy that gradually expands the alignment interval during training. This process induces a contraction mapping, which improves convergence and training stability. Empirically, ISA-DRE achieves competitive accuracy with significantly fewer function evaluations compared to prior methods, resulting in much faster inference and making it well suited for real-time and interactive applications.