Weichen Zhao

Weichen Zhao, Chenguang Wang, Congying Han et al.

The over-smoothing problem is an obstacle of developing deep graph neural network (GNN). Although many approaches to improve the over-smoothing problem have been proposed, there is still a lack of comprehensive understanding and conclusion of this problem. In this work, we analyze the over-smoothing problem from the Markov chain perspective. We focus on message passing of GNN and first establish a connection between GNNs and Markov chains on the graph. GNNs are divided into two classes of operator-consistent and operator-inconsistent based on whether the corresponding Markov chains are time-homogeneous. Next we attribute the over-smoothing problem to the convergence of an arbitrary initial distribution to a stationary distribution. Based on this, we prove that although the previously proposed methods can alleviate over-smoothing, but these methods cannot avoid the over-smoothing problem. In addition, we give the conclusion of the over-smoothing problem in two types of GNNs in the Markovian sense. On the one hand, operator-consistent GNN cannot avoid over-smoothing at an exponential rate. On the other hand, operator-inconsistent GNN is not always over-smoothing. Further, we investigate the existence of the limiting distribution of the time-inhomogeneous Markov chain, from which we derive a sufficient condition for operator-inconsistent GNN to avoid over-smoothing. Finally, we design experiments to verify our findings. Results show that our proposed sufficient condition can effectively improve over-smoothing problem in operator-inconsistent GNN and enhance the performance of the model.

Qianhan Zeng, Yingqiu Zhu, Xuening Zhu et al.

Labeling mistakes are frequently encountered in real-world applications. If not treated well, the labeling mistakes can deteriorate the classification performances of a model seriously. To address this issue, we propose an improved Naive Bayes method for text classification. It is analytically simple and free of subjective judgements on the correct and incorrect labels. By specifying the generating mechanism of incorrect labels, we optimize the corresponding log-likelihood function iteratively by using an EM algorithm. Our simulation and experiment results show that the improved Naive Bayes method greatly improves the performances of the Naive Bayes method with mislabeled data.

Kaiyuan Cui, Xinyan Wang, Zicheng Zhang et al.

Continuous graph neural models based on differential equations have expanded the architecture of graph neural networks (GNNs). Due to the connection between graph diffusion and message passing, diffusion-based models have been widely studied. However, diffusion naturally drives the system towards an equilibrium state, leading to issues like over-smoothing. To this end, we propose GRADE inspired by graph aggregation-diffusion equations, which includes the delicate balance between nonlinear diffusion and aggregation induced by interaction potentials. The node representations obtained through aggregation-diffusion equations exhibit metastability, indicating that features can aggregate into multiple clusters. In addition, the dynamics within these clusters can persist for long time periods, offering the potential to alleviate over-smoothing effects. This nonlinear diffusion in our model generalizes existing diffusion-based models and establishes a connection with classical GNNs. We prove that GRADE achieves competitive performance across various benchmarks and alleviates the over-smoothing issue in GNNs evidenced by the enhanced Dirichlet energy.

Weichen Zhao, Chenguang Wang, Xinyan Wang et al.

This paper presents an analytical study of the oversmoothing issue in diffusion-based Graph Neural Networks (GNNs). Generalizing beyond extant approaches grounded in random walk analysis or particle systems, we approach this problem through operator semigroup theory. This theoretical framework allows us to rigorously prove that oversmoothing is intrinsically linked to the ergodicity of the diffusion operator. Relying on semigroup method, we can quantitatively analyze the dynamic of graph diffusion and give a specific mathematical form of the smoothing feature by ergodicity and invariant measure of operator, which improves previous works only show existence of oversmoothing. This finding further poses a general and mild ergodicity-breaking condition, encompassing the various specific solutions previously offered, thereby presenting a more universal and theoretically grounded approach to relieve oversmoothing in diffusion-based GNNs. Additionally, we offer a probabilistic interpretation of our theory, forging a link with prior works and broadening the theoretical horizon. Our experimental results reveal that this ergodicity-breaking term effectively mitigates oversmoothing measured by Dirichlet energy, and simultaneously enhances performance in node classification tasks.

Zhengyang Wei, Weichen Zhao, Chang Liu

This work analyzes accelerating and decelerating wall-driven flows by quantifying the upper bound of transient energy growth using a Lyapunov-type approach. By formulating the linearized Navier-Stokes equations as a linear time-varying system and constructing a time-dependent Lyapunov function, we obtain an upper bound on transient energy growth by solving linear matrix inequalities. This Lyapunov method can obtain the upper bound of transient energy growth that closely matches transient growth computed via the singular value decomposition of the state-transition matrix of linear time-varying systems. Our analysis captures that decelerating base flows exhibit significantly larger transient growth compared with accelerating flows. Our Lyapunov method offers the advantages of providing a certificate of uniform stability and an invariant set to bound the solution trajectory.

Yuxing Lu, J. Ben Tamo, Weichen Zhao et al.

Large language models are strong sequence predictors, yet standard inference relies on immutable context histories. After making an error at generation step t, the model lacks an updatable memory mechanism that improves predictions for step t+1. We propose LLM-as-RNN, an inference-only framework that turns a frozen LLM into a recurrent predictor by representing its hidden state as natural-language memory. This state, implemented as a structured system-prompt summary, is updated at each timestep via feedback-driven text rewrites, enabling learning without parameter updates. Under a fixed token budget, LLM-as-RNN corrects errors and retains task-relevant patterns, effectively performing online learning through language. We evaluate the method on three sequential benchmarks in healthcare, meteorology, and finance across Llama, Gemma, and GPT model families. LLM-as-RNN significantly outperforms zero-shot, full-history, and MemPrompt baselines, improving predictive accuracy by 6.5% on average, while producing interpretable, human-readable learning traces absent in standard context accumulation.

Chenguang Wang, Xiaoyu Zhang, Kaiyuan Cui et al.

Training neural samplers directly from unnormalized densities without access to target distribution samples presents a significant challenge. A critical desideratum in these settings is achieving comprehensive mode coverage, ensuring the sampler captures the full diversity of the target distribution. However, prevailing methods often circumvent the lack of target data by optimizing reverse KL-based objectives. Such objectives inherently exhibit mode-seeking behavior, potentially leading to incomplete representation of the underlying distribution. While alternative approaches strive for better mode coverage, they typically rely on implicit mechanisms like heuristics or iterative refinement. In this work, we propose a principled approach for training diffusion-based samplers by directly targeting an objective analogous to the forward KL divergence, which is conceptually known to encourage mode coverage. We introduce \textit{Importance Weighted Score Matching}, a method that optimizes this desired mode-covering objective by re-weighting the score matching loss using tractable importance sampling estimates, thereby overcoming the absence of target distribution data. We also provide theoretical analysis of the bias and variance for our proposed Monte Carlo estimator and the practical loss function used in our method. Experiments on increasingly complex multi-modal distributions, including 2D Gaussian Mixture Models with up to 120 modes and challenging particle systems with inherent symmetries -- demonstrate that our approach consistently outperforms existing neural samplers across all distributional distance metrics, achieving state-of-the-art results on all benchmarks.