Yang Cao

Yang Cao, Yubin Chen, Xuyang Guo et al.

The rapid advancement of large language models (LLMs) has led to significant breakthroughs in automated mathematical reasoning and scientific discovery. Georgiev, G${ó}$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.

Yang Cao, Sikun Yang, Yujiu Yang et al.

Two-step approaches combining pre-trained large language model embeddings and anomaly detectors demonstrate strong performance in text anomaly detection by leveraging rich semantic representations. However, high-dimensional dense embeddings extracted by large language models pose challenges due to substantial memory requirements and high computation time. To address this challenge, we introduce the Simplified Isolation Kernel (SIK), which maps high-dimensional dense embeddings to lower-dimensional sparse representations while preserving crucial anomaly characteristics. SIK has linear time complexity and significantly reduces space complexity through its innovative boundary-focused feature mapping. Experiments across 7 datasets demonstrate that SIK achieves better detection performance than 11 state-of-the-art (SOTA) anomaly detection algorithms while maintaining computational efficiency and low memory cost. All code and demonstrations are available at https://github.com/charles-cao/SIK.

Yang Cao, Haolong Xiang, Hang Zhang et al.

Anomaly detection is a longstanding and active research area that has many applications in domains such as finance, security, and manufacturing. However, the efficiency and performance of anomaly detection algorithms are challenged by the large-scale, high-dimensional, and heterogeneous data that are prevalent in the era of big data. Isolation-based unsupervised anomaly detection is a novel and effective approach for identifying anomalies in data. It relies on the idea that anomalies are few and different from normal instances, and thus can be easily isolated by random partitioning. Isolation-based methods have several advantages over existing methods, such as low computational complexity, low memory usage, high scalability, robustness to noise and irrelevant features, and no need for prior knowledge or heavy parameter tuning. In this survey, we review the state-of-the-art isolation-based anomaly detection methods, including their data partitioning strategies, anomaly score functions, and algorithmic details. We also discuss some extensions and applications of isolation-based methods in different scenarios, such as detecting anomalies in streaming data, time series, trajectory, and image datasets. Finally, we identify some open challenges and future directions for isolation-based anomaly detection research.

Yang Cao, Sikun Yang, Chen Li et al.

Text anomaly detection is crucial for identifying spam, misinformation, and offensive language in natural language processing tasks. Despite the growing adoption of embedding-based methods, their effectiveness and generalizability across diverse application scenarios remain under-explored. To address this, we present TAD-Bench, a comprehensive benchmark designed to systematically evaluate embedding-based approaches for text anomaly detection. TAD-Bench integrates multiple datasets spanning different domains, combining state-of-the-art embeddings from large language models with a variety of anomaly detection algorithms. Through extensive experiments, we analyze the interplay between embeddings and detection methods, uncovering their strengths, weaknesses, and applicability to different tasks. These findings offer new perspectives on building more robust, efficient, and generalizable anomaly detection systems for real-world applications.