Muon in Associative Memory Learning: Training Dynamics and Scaling Laws

Muon updates matrix parameters via the matrix sign of the gradient and has shown strong empirical gains, yet its dynamics and scaling behavior remain unclear in theory. We study Muon in a linear associative memory model with softmax retrieval and a hierarchical frequency spectrum over query-answer pairs, with and without label noise. In this setting, we show that Gradient Descent (GD) learns frequency components at highly imbalanced rates, leading to slow convergence bottlenecked by low-frequency components. In contrast, the Muon optimizer mitigates this imbalance, leading to faster and more uniform progress. Specifically, in the noiseless case, Muon achieves an exponential speedup over GD; in the noisy case with a power-decay frequency spectrum, we derive Muon's optimization scaling law and demonstrate its superior scaling efficiency over GD. Furthermore, we show that Muon can be interpreted as an implicit matrix preconditioner arising from adaptive task alignment and block-symmetric gradient structure. In contrast, the preconditioner with coordinate-wise sign operator could match Muon under oracle access to unknown task representations, which is infeasible for SignGD in practice. Experiments on synthetic long-tail classification and LLaMA-style pre-training corroborate the theory.

Towards Solving the Gilbert-Pollak Conjecture via Large Language Models

The Gilbert-Pollak Conjecture \citep{gilbert1968steiner}, also known as the Steiner Ratio Conjecture, states that for any finite point set in the Euclidean plane, the Steiner minimum tree has length at least $\sqrt{3}/2 \approx 0.866$ times that of the Euclidean minimum spanning tree (the Steiner ratio). A sequence of improvements through the 1980s culminated in a lower bound of $0.824$, with no substantial progress reported over the past three decades. Recent advances in LLMs have demonstrated strong performance on contest-level mathematical problems, yet their potential for addressing open, research-level questions remains largely unexplored. In this work, we present a novel AI system for obtaining tighter lower bounds on the Steiner ratio. Rather than directly prompting LLMs to solve the conjecture, we task them with generating rule-constrained geometric lemmas implemented as executable code. These lemmas are then used to construct a collection of specialized functions, which we call verification functions, that yield theoretically certified lower bounds of the Steiner ratio. Through progressive lemma refinement driven by reflection, the system establishes a new certified lower bound of 0.8559 for the Steiner ratio. The entire research effort involves only thousands of LLM calls, demonstrating the strong potential of LLM-based systems for advanced mathematical research.

Efficient-VLN: A Training-Efficient Vision-Language Navigation Model

Multimodal large language models (MLLMs) have shown promising potential in Vision-Language Navigation (VLN). However, their practical development is severely hindered by the substantial training overhead. We recognize two key issues that contribute to the overhead: (1) the quadratic computational burden from processing long-horizon historical observations as massive sequences of tokens, and (2) the exploration-efficiency trade-off in DAgger, i.e., a data aggregation process of collecting agent-explored trajectories. While more exploration yields effective error-recovery trajectories for handling test-time distribution shifts, it comes at the cost of longer trajectory lengths for both training and inference. To address these challenges, we propose Efficient-VLN, a training-efficient VLN model. Specifically, to mitigate the token processing burden, we design two efficient memory mechanisms: a progressive memory that dynamically allocates more tokens to recent observations, and a learnable recursive memory that utilizes the key-value cache of learnable tokens as the memory state. Moreover, we introduce a dynamic mixed policy to balance the exploration-efficiency trade-off. Extensive experiments show that Efficient-VLN achieves state-of-the-art performance on R2R-CE (64.2% SR) and RxR-CE (67.0% SR). Critically, our model consumes merely 282 H800 GPU hours, demonstrating a dramatic reduction in training overhead compared to state-of-the-art methods.

Rethinking Chain-of-Thought Reasoning for Videos

Chain-of-thought (CoT) reasoning has been highly successful in solving complex tasks in natural language processing, and recent multimodal large language models (MLLMs) have extended this paradigm to video reasoning. However, these models typically build on lengthy reasoning chains and large numbers of input visual tokens. Motivated by empirical observations from our benchmark study, we hypothesize that concise reasoning combined with a reduced set of visual tokens can be sufficient for effective video reasoning. To evaluate this hypothesis, we design and validate an efficient post-training and inference framework that enhances a video MLLM's reasoning capability. Our framework enables models to operate on compressed visual tokens and generate brief reasoning traces prior to answering. The resulting models achieve substantially improved inference efficiency, deliver competitive performance across diverse benchmarks, and avoid reliance on manual CoT annotations or supervised fine-tuning. Collectively, our results suggest that long, human-like CoT reasoning may not be necessary for general video reasoning, and that concise reasoning can be both effective and efficient. Our code will be released at https://github.com/LaVi-Lab/Rethink_CoT_Video.

Distribution Matching Variational AutoEncoder

Most visual generative models compress images into a latent space before applying diffusion or autoregressive modelling. Yet, existing approaches such as VAEs and foundation model aligned encoders implicitly constrain the latent space without explicitly shaping its distribution, making it unclear which types of distributions are optimal for modeling. We introduce \textbf{Distribution-Matching VAE} (\textbf{DMVAE}), which explicitly aligns the encoder's latent distribution with an arbitrary reference distribution via a distribution matching constraint. This generalizes beyond the Gaussian prior of conventional VAEs, enabling alignment with distributions derived from self-supervised features, diffusion noise, or other prior distributions. With DMVAE, we can systematically investigate which latent distributions are more conducive to modeling, and we find that SSL-derived distributions provide an excellent balance between reconstruction fidelity and modeling efficiency, reaching gFID equals 3.2 on ImageNet with only 64 training epochs. Our results suggest that choosing a suitable latent distribution structure (achieved via distribution-level alignment), rather than relying on fixed priors, is key to bridging the gap between easy-to-model latents and high-fidelity image synthesis. Code is avaliable at https://github.com/sen-ye/dmvae.

Hierarchical Federated Learning for Social Network with Mobility

Federated Learning (FL) offers a decentralized solution that allows collaborative local model training and global aggregation, thereby protecting data privacy. In conventional FL frameworks, data privacy is typically preserved under the assumption that local data remains absolutely private, whereas the mobility of clients is frequently neglected in explicit modeling. In this paper, we propose a hierarchical federated learning framework based on the social network with mobility namely HFL-SNM that considers both data sharing among clients and their mobility patterns. Under the constraints of limited resources, we formulate a joint optimization problem of resource allocation and client scheduling, which objective is to minimize the energy consumption of clients during the FL process. In social network, we introduce the concepts of Effective Data Coverage Rate and Redundant Data Coverage Rate. We analyze the impact of effective data and redundant data on the model performance through preliminary experiments. We decouple the optimization problem into multiple sub-problems, analyze them based on preliminary experimental results, and propose Dynamic Optimization in Social Network with Mobility (DO-SNM) algorithm. Experimental results demonstrate that our algorithm achieves superior model performance while significantly reducing energy consumption, compared to traditional baseline algorithms.

Solving the Hubbard model with Neural Quantum States

The rapid development of neural quantum states (NQS) has established it as a promising framework for studying quantum many-body systems. In this work, by leveraging the cutting-edge transformer-based architectures and developing highly efficient optimization algorithms, we achieve the state-of-the-art results for the doped two-dimensional (2D) Hubbard model, arguably the minimum model for high-Tc superconductivity. Interestingly, we find different attention heads in the NQS ansatz can directly encode correlations at different scales, making it capable of capturing long-range correlations and entanglements in strongly correlated systems. With these advances, we establish the half-filled stripe in the ground state of 2D Hubbard model with the next nearest neighboring hoppings, consistent with experimental observations in cuprates. Our work establishes NQS as a powerful tool for solving challenging many-fermions systems.

Diagnosing and Improving Diffusion Models by Estimating the Optimal Loss Value

Diffusion models have achieved remarkable success in generative modeling. Despite more stable training, the loss of diffusion models is not indicative of absolute data-fitting quality, since its optimal value is typically not zero but unknown, leading to confusion between large optimal loss and insufficient model capacity. In this work, we advocate the need to estimate the optimal loss value for diagnosing and improving diffusion models. We first derive the optimal loss in closed form under a unified formulation of diffusion models, and develop effective estimators for it, including a stochastic variant scalable to large datasets with proper control of variance and bias. With this tool, we unlock the inherent metric for diagnosing the training quality of mainstream diffusion model variants, and develop a more performant training schedule based on the optimal loss. Moreover, using models with 120M to 1.5B parameters, we find that the power law is better demonstrated after subtracting the optimal loss from the actual training loss, suggesting a more principled setting for investigating the scaling law for diffusion models.

The Sample Complexity of Online Strategic Decision Making with Information Asymmetry and Knowledge Transportability

Information asymmetry is a pervasive feature of multi-agent systems, especially evident in economics and social sciences. In these settings, agents tailor their actions based on private information to maximize their rewards. These strategic behaviors often introduce complexities due to confounding variables. Simultaneously, knowledge transportability poses another significant challenge, arising from the difficulties of conducting experiments in target environments. It requires transferring knowledge from environments where empirical data is more readily available. Against these backdrops, this paper explores a fundamental question in online learning: Can we employ non-i.i.d. actions to learn about confounders even when requiring knowledge transfer? We present a sample-efficient algorithm designed to accurately identify system dynamics under information asymmetry and to navigate the challenges of knowledge transfer effectively in reinforcement learning, framed within an online strategic interaction model. Our method provably achieves learning of an $\epsilon$-optimal policy with a tight sample complexity of $O(1/\epsilon^2)$.

DeformCL: Learning Deformable Centerline Representation for Vessel Extraction in 3D Medical Image

In the field of 3D medical imaging, accurately extracting and representing the blood vessels with curvilinear structures holds paramount importance for clinical diagnosis. Previous methods have commonly relied on discrete representation like mask, often resulting in local fractures or scattered fragments due to the inherent limitations of the per-pixel classification paradigm. In this work, we introduce DeformCL, a new continuous representation based on Deformable Centerlines, where centerline points act as nodes connected by edges that capture spatial relationships. Compared with previous representations, DeformCL offers three key advantages: natural connectivity, noise robustness, and interaction facility. We present a comprehensive training pipeline structured in a cascaded manner to fully exploit these favorable properties of DeformCL. Extensive experiments on four 3D vessel segmentation datasets demonstrate the effectiveness and superiority of our method. Furthermore, the visualization of curved planar reformation images validates the clinical significance of the proposed framework. We release the code in https://github.com/barry664/DeformCL