arXiv subjects
Publications and source records attributed to Song Yue.
Marton's covering conjecture studies finite sets in high-dimensional binary spaces whose pairwise sums create relatively few new elements. It predicts that every such set can be described efficiently by shifted copies of one linear subspace of comparable size. Gowers, Green, Manners, and Tao [GGMT25] proved the conjecture with exponent $12$. Liao [L24] improved the exponent to $9$. We improve it further to $8.873$.
The rapid growth of AI conference submissions has created an overwhelming reviewing burden. To alleviate this, recent venues such as ICLR 2026 introduced a reviewer nomination policy: each submission must nominate one of its authors as a reviewer, and any paper nominating an irresponsible reviewer is desk-rejected. We study this new policy from the perspective of author welfare. Assuming each author carries a probability of being irresponsible, we ask: how can authors (or automated systems) nominate reviewers to minimize the risk of desk rejections? We formalize and analyze three variants of the desk-rejection risk minimization problem. The basic problem, which minimizes expected desk rejections, is solved optimally by a simple greedy algorithm. We then introduce hard and soft nomination limit variants that constrain how many papers may nominate the same author, preventing widespread failures if one author is irresponsible. These formulations connect to classical optimization frameworks, including minimum-cost flow and linear programming, allowing us to design efficient, principled nomination strategies. Our results provide the first theoretical study for reviewer nomination policies, offering both conceptual insights and practical directions for authors to wisely choose which co-author should serve as the nominated reciprocal reviewer.
The rapid advancement of large language models (LLMs) has led to significant breakthroughs in automated mathematical reasoning and scientific discovery. Georgiev, G${\'o}$mez-Serrano, Tao, and Wagner [GGSTW+25] demonstrate that AI systems can explore new constructions and improve existing bounds, illustrating the growing potential of LLMs to accelerate mathematical discovery. Similarly, Bubeck et al. [BCE+25] show that GPT-5 can meaningfully contribute to scientific workflows, from proposing hypotheses to generating proofs and analyses. Despite these advances, a rigorous evaluation of these models on canonical, graduate-level mathematical theory remains necessary to understand their baseline reasoning capabilities. In this paper, we present a comprehensive benchmark of four frontier models: GPT-5-Thinking, Gemini-3-Pro, Claude-Sonnet-4.5-Thinking, and Grok-4 against the classic curriculum of Randomized Algorithms by Motwani and Raghavan [MR95]. We tasked each model with generating formal LaTeX proofs for a series of lemmas and exercises spanning the textbook. We find that while the top-tier models (Gemini, and Claude) achieve a high accuracy rate (approx. 66%), demonstrating a robust grasp of probabilistic method and formal logic, other models lag significantly in consistency (approx. 40%). We provide a qualitative analysis of the generated proofs, highlighting differences in conciseness, hallucination rates, and logical structure. Our results suggest that while frontier models have reached a threshold of proficiency suitable for graduate-level pedagogical assistance and formalization, significant variance exists in their reliability for rigorous mathematical derivation. The code and the full set of LLM-generated responses are open-sourced and publicly available at https://github.com/magiclinux/math_benchmark_probability.
Large Reasoning Models (LRMs) have become a central focus in today's large language model (LLM) research, where models are designed to output a step-by-step thinking process before arriving at a final answer to handle complex reasoning tasks. Despite their promise, recent empirical studies (e.g., [Shojaee et al., 2025] from Apple) suggest that this thinking process may not actually enhance reasoning ability, where LLMs without explicit reasoning actually outperform LRMs on tasks with low or high complexity. In this work, we revisit these findings and investigate whether the limitations of LRMs persist when tool augmentations are introduced. We incorporate two types of tools, Python interpreters and scratchpads, and evaluate three representative LLMs and their LRM counterparts on Apple's benchmark reasoning puzzles. Our results show that, with proper tool use, LRMs consistently outperform their non-reasoning counterparts across all levels of task complexity. These findings challenge the recent narrative that reasoning is an illusion and highlight the potential of tool-augmented LRMs for solving complex problems.
Large Language Models (LLMs) exhibit exceptional proficiency in handling extensive context windows in natural language. Nevertheless, the quadratic scaling of attention computation relative to sequence length creates substantial efficiency bottlenecks, necessitating the development of I/O-optimized algorithms. In this work, we conduct a systematic examination of the I/O complexity inherent in attention mechanisms, with a specific emphasis on the backward pass under both small and large cache settings. By leveraging the red-blue pebble game framework, we derive tight bounds for I/O complexity across the full spectrum of cache sizes. We validate that FlashAttention, one of the current industry standards, achieves optimality in the large-cache scenario for both forward and backward passes. Conversely, for small-cache environments, we introduce a novel algorithm that outperforms contemporary methods and successfully attains theoretical tight bounds. Furthermore, we expand our investigation to include sparse attention by establishing granular lower bounds for both forward and backward passes across all cache configurations. Ultimately, our results solidify the theoretical framework regarding I/O complexity in attention mechanisms, providing critical guidance for the development of efficient LLM training and inference systems.
Stochastic gradient descent (SGD) now acts as a fundamental part of optimization in current machine learning. Meanwhile, deep learning architectures have shown outstanding performance in a wide range of fields, such as natural language processing, bioinformatics, and computer vision. Nevertheless, as the parameter size $d$ increases, these models encounter serious efficiency challenges. Previous studies show that the per step calculation expense scales linearly with the input size $d$. To mitigate this, our paper explores inherent patterns, such as Kronecker products within the training examples. We consider input data points that can be represented as tensor products of lower-dimensional vectors. We introduce a novel stochastic optimization method where the computational load for every update scales sublinearly with $d$, assuming moderate structural properties of the inputs. We believe our research is the first work achieving this result, representing a significant step forward for efficient deep learning optimization. Our theoretical findings are supported by a formal theorem, demonstrating that the proposed algorithm can train a two-layer fully connected neural network with a per-iteration cost independent of $d$.
We report a comparative study of the specific heat, electrical resistivity and thermal conductivity of the quasi-two-dimensional purple bronzes Na$_{0.9}$Mo$_6$O$_{17}$ and K$_{0.9}$Mo$_6$O$_{17}$, with special emphasis on the behavior near their respective charge-density-wave transition temperatures $T_P$. The contrasting behavior of both the transport and the thermodynamic properties near $T_P$ is argued to arise predominantly from the different levels of intrinsic disorder in the two systems. A significant proportion of the enhancement of the thermal conductivity above $T_P$ in Na$_{0.9}$Mo$_6$O$_{17}$, and to a lesser extent in K$_{0.9}$Mo$_6$O$_{17}$, is attributed to the emergence of phason excitations.