Xudong Wang

Chenghao Li, Chaoning Zhang, Yi Lu et al.

Chain-of-Thought (CoT) prompting enhances the reasoning of large language models (LLMs) by decomposing problems into sequential steps, mimicking human logic and reducing errors. However, complex tasks with vast solution spaces and vague constraints often exceed the capacity of a single reasoning chain. Inspired by Minimal Free Resolution (MFR) in commutative algebra and algebraic geometry, we propose Syzygy of Thoughts (SoT)-a novel framework that extends CoT by introducing auxiliary, interrelated reasoning paths. SoT captures deeper logical dependencies, enabling more robust and structured problem-solving. MFR decomposes a module into a sequence of free modules with minimal rank, providing a structured analytical approach to complex systems. This method introduces the concepts of "Module", "Betti numbers","Freeness", "Mapping", "Exactness" and "Minimality", enabling the systematic decomposition of the original complex problem into logically complete minimal subproblems while preserving key problem features and reducing reasoning length. We tested SoT across diverse datasets (e.g., GSM8K, MATH) and models (e.g., GPT-4o-mini, Qwen2.5), achieving inference accuracy that matches or surpasses mainstream CoTs standards. Additionally, by aligning the sampling process with algebraic constraints, our approach enhances the scalability of inference time in LLMs, ensuring both transparent reasoning and high performance. Our code will be publicly available at https://github.com/dlMARiA/Syzygy-of-thoughts.

Qing Wu, Xu Guo, Lixuan Chen et al.

X-ray CT often suffers from shadowing and streaking artifacts in the presence of metallic materials, which severely degrade imaging quality. Physically, the linear attenuation coefficients (LACs) of metals vary significantly with X-ray energy, causing a nonlinear beam hardening effect (BHE) in CT measurements. Reconstructing CT images from metal-corrupted measurements consequently becomes a challenging nonlinear inverse problem. Existing state-of-the-art (SOTA) metal artifact reduction (MAR) algorithms rely on supervised learning with numerous paired CT samples. While promising, these supervised methods often assume that the unknown LACs are energy-independent, ignoring the energy-induced BHE, which results in limited generalization. Moreover, the requirement for large datasets also limits their applications in real-world scenarios. In this work, we propose Density neural representation (Diner), a novel unsupervised MAR method. Our key innovation lies in formulating MAR as an energy-independent density reconstruction problem that strictly adheres to the photon-tissue absorption physical model. This model is inherently nonlinear and complex, making it a rarely considered approach in inverse imaging problems. By introducing the water-equivalent tissues approximation and a new polychromatic model to characterize the nonlinear CT acquisition process, we directly learn the neural representation of the density map from raw measurements without using external training data. This energy-independent density reconstruction framework fundamentally resolves the nonlinear BHE, enabling superior MAR performance across a wide range of scanning scenarios. Extensive experiments on both simulated and real-world datasets demonstrate the superiority of our unsupervised Diner over popular supervised methods in terms of MAR performance and robustness.