A NALYTICALD ISCOVERY OF M ANIFOLD WITH M A-CHINE L EARNING Y afei Shen 1 & Huan-Fei Ma 1, & Ling Y ang 1, 1 School of Mathematical Sciences, Soochow University, Suzhou 215006, China A BSTRACT Understanding low-dimensional structures within high-dimensional data is crucial for visualization, interpretation, and denoising in complex datasets. Despite the advancements in manifold learning techniques, key challenges--such as limited global insight and the lack of interpretable analytical descriptions--remain unresolved. In this work, we introduce a novel framework, GAMLA (Global Analytical Manifold Learning using Auto-encoding). GAMLA employs a two-round training process within an auto-encoding framework to derive both character and complementary representations for the underlying manifold. With the character representation, the manifold is represented by a parametric function which unfold the manifold to provide a global coordinate. While with the complementary representation, an approximate explicit manifold description is developed, offering a global and analytical representation of smooth manifolds underlying high-dimensional datasets. This enables the analytical derivation of geometric properties such as curvature and normal vectors. Moreover, we find the two representations together decompose the whole latent space and can thus characterize the local spatial structure surrounding the manifold, proving particularly effective in anomaly detection and categorization. Through extensive experiments on benchmark datasets and real-world applications, GAMLA demonstrates its ability to achieve computational efficiency and interpretability while providing precise geometric and structural insights. This framework bridges the gap between data-driven manifold learning and analytical geometry, presenting a versatile tool for exploring the intrinsic properties of complex data sets. 1 I NTRODUCTION Discovering low-dimensional structures, particularly their geometric properties, from high-dimensional data clouds enables visualization, denoising, and interpretation of complex datasets (Meil a & Zhang, 2023; Belkin & Niyogi, 2003; van der Maaten & Hinton, 2008; McInnes & Healy, 2018; Luo & Hu, 2020). As a result, the concept of manifold learning has attracted significant attention, leading to numerous breakthroughs over the past two decades.

Verification is crucial for effective mathematical reasoning. We present a new temporal consistency method where verifiers iteratively refine their judgments based on the previous assessment. Unlike one-round verification or multi-model debate approaches, our method leverages consistency in a sequence of self-reflection actions to improve verification accuracy. Empirical evaluations across diverse mathematical process error identification benchmarks (Mathcheck, ProcessBench, and PRM800K) show consistent performance improvements over baseline methods. When applied to the recent DeepSeek R1 distilled models, our method demonstrates strong performance, enabling 7B/8B distilled models to outperform all 70B/72B models and GPT-4o on ProcessBench. Notably, the distilled 14B model with our method achieves performance comparable to Deepseek-R1. Our codes are available at https://github.com/jcguo123/Temporal-Consistency

We present that hierarchical LLM reasoning via scaling thought templates can effectively optimize the reasoning search space and outperform the mathematical reasoning capabilities of powerful LLMs like OpenAI o1-preview and DeepSeek V3. We train our ReasonFlux-32B model with only 8 GPUs and introduces three innovations: (i) a structured and generic thought template library, containing around 500 high-level thought templates capable of generalizing to similar or relevant reasoning problems; (ii) performing hierarchical reinforcement learning on a sequence of thought templates instead of original long CoT data, optimizing a base LLM to plan out an optimal template trajectory for gradually handling complex problems; (iii) a brand new inference scaling system that enables hierarchical LLM reasoning by adaptively scaling thought templates at inference time. With a template trajectory containing sequential thought templates, our ReasonFlux-32B significantly advances math reasoning capabilities to state-of-the-art levels. Notably, on the MATH benchmark, it achieves an accuracy of 91.2% and surpasses o1-preview by 6.7%. On the USA Math Olympiad (AIME) benchmark, ReasonFlux-32B solves an average of 56.7% of problems, surpassing o1-preview and DeepSeek-V3 by 27% and 45%, respectively.

Recent research has leveraged large language model multi-agent systems for complex problem-solving while trying to reduce the manual effort required to build them, driving the development of automated agent workflow optimization methods. However, existing methods remain inflexible due to representational limitations, a lack of adaptability, and poor scalability when relying on discrete optimization techniques. We address these challenges with ScoreFlow, a simple yet high-performance framework that leverages efficient gradient-based optimization in a continuous space. ScoreFlow incorporates Score-DPO, a novel variant of the direct preference optimization method that accounts for quantitative feedback. Across six benchmarks spanning question answering, coding, and mathematical reasoning, ScoreFlow achieves an 8.2% improvement over existing baselines. Moreover, it empowers smaller models to outperform larger ones with lower inference costs. Project: https://github.com/Gen-Verse/ScoreFlow

In recent years, Graph Neural Networks (GNNs) have achieved remarkable success in many graph mining tasks. However, scaling them to large graphs is challenging due to the high computational and storage costs of repeated feature propagation and non-linear transformation during training. One commonly employed approach to address this challenge is model-simplification, which only executes the Propagation (P) once in the pre-processing, and Combine (C) these receptive fields in different ways and then feed them into a simple model for better performance. Despite their high predictive performance and scalability, these methods still face two limitations. First, existing approaches mainly focus on exploring different C methods from the model perspective, neglecting the crucial problem of performance degradation with increasing P depth from the data-centric perspective, known as the over-smoothing problem. Second, pre-processing overhead takes up most of the end-to-end processing time, especially for large-scale graphs. To address these limitations, we present random walk with noise masking (RMask), a plug-and-play module compatible with the existing model-simplification works. This module enables the exploration of deeper GNNs while preserving their scalability. Unlike the previous model-simplification works, we focus on continuous P and found that the noise existing inside each P is the cause of the over-smoothing issue, and use the efficient masking mechanism to eliminate them. Experimental results on six real-world datasets demonstrate that model-simplification works equipped with RMask yield superior performance compared to their original version and can make a good trade-off between accuracy and efficiency.

Efficient training of large-scale heterogeneous graphs is of paramount importance in real-world applications. However, existing approaches typically explore simplified models to mitigate resource and time overhead, neglecting the crucial aspect of simplifying large-scale heterogeneous graphs from the data-centric perspective. Addressing this gap, HGCond introduces graph condensation (GC) in heterogeneous graphs and generates a small condensed graph for efficient model training. Despite its efficacy in graph generation, HGCond encounters two significant limitations. The first is low effectiveness, HGCond excessively relies on the simplest relay model for the condensation procedure, which restricts the ability to exert powerful Heterogeneous Graph Neural Networks (HGNNs) with flexible condensation ratio and limits the generalization ability. The second is low efficiency, HGCond follows the existing GC methods designed for homogeneous graphs and leverages the sophisticated optimization paradigm, resulting in a time-consuming condensing procedure. In light of these challenges, we present the first Training \underline{Free} Heterogeneous Graph Condensation method, termed FreeHGC, facilitating both efficient and high-quality generation of heterogeneous condensed graphs. Specifically, we reformulate the heterogeneous graph condensation problem as a data selection issue, offering a new perspective for assessing and condensing representative nodes and edges in the heterogeneous graphs. By leveraging rich meta-paths, we introduce a new, high-quality heterogeneous data selection criterion to select target-type nodes. Furthermore, two training-free condensation strategies for heterogeneous graphs are designed to condense and synthesize other-types nodes effectively.

While time series diffusion models have received considerable focus from many recent works, the performance of existing models remains highly unstable. Factors limiting time series diffusion models include insufficient time series datasets and the absence of guidance. To address these limitations, we propose a Retrieval- Augmented Time series Diffusion model (RATD). The framework of RATD consists of two parts: an embedding-based retrieval process and a reference-guided diffusion model. In the first part, RATD retrieves the time series that are most relevant to historical time series from the database as references. The references are utilized to guide the denoising process in the second part. Our approach allows leveraging meaningful samples within the database to aid in sampling, thus maximizing the utilization of datasets. Meanwhile, this reference-guided mechanism also compensates for the deficiencies of existing time series diffusion models in terms of guidance. Experiments and visualizations on multiple datasets demonstrate the effectiveness of our approach, particularly in complicated prediction tasks.

Efficient data selection is crucial to accelerate the pretraining of large language models (LLMs). While various methods have been proposed to enhance data efficiency, limited research has addressed the inherent conflicts between these approaches to achieve optimal data selection for LLM pretraining. To tackle this problem, we propose a novel multi-agent collaborative data selection mechanism. In this framework, each data selection method serves as an independent agent, and an agent console is designed to dynamically integrate the information from all agents throughout the LLM training process. We conduct extensive empirical studies to evaluate our multi-agent framework. The experimental results demonstrate that our approach significantly improves data efficiency, accelerates convergence in LLM training, and achieves an average performance gain up to 10.5% across multiple language model benchmarks compared to the state-of-the-art methods.

Large language models (LLMs) like GPT-4, PaLM, and LLaMA have shown significant improvements in various reasoning tasks. However, smaller models such as Llama-3-8B and DeepSeekMath-Base still struggle with complex mathematical reasoning because they fail to effectively identify and correct reasoning errors. Recent reflection-based methods aim to address these issues by enabling self-reflection and self-correction, but they still face challenges in independently detecting errors in their reasoning steps. To overcome these limitations, we propose SuperCorrect, a novel two-stage framework that uses a large teacher model to supervise and correct both the reasoning and reflection processes of a smaller student model. In the first stage, we extract hierarchical high-level and detailed thought templates from the teacher model to guide the student model in eliciting more fine-grained reasoning thoughts. In the second stage, we introduce cross-model collaborative direct preference optimization (DPO) to enhance the self-correction abilities of the student model by following the teacher's correction traces during training. This cross-model DPO approach teaches the student model to effectively locate and resolve erroneous thoughts with error-driven insights from the teacher model, breaking the bottleneck of its thoughts and acquiring new skills and knowledge to tackle challenging problems. Extensive experiments consistently demonstrate our superiority over previous methods. Notably, our SuperCorrect-7B model significantly surpasses powerful DeepSeekMath-7B by 7.8%/5.3% and Qwen2.5-Math-7B by 15.1%/6.3% on MATH/GSM8K benchmarks, achieving new SOTA performance among all 7B models. Code: https://github.com/YangLing0818/SuperCorrect-llm

We introduce Buffer of Thoughts (BoT), a novel and versatile thought-augmented reasoning approach for enhancing accuracy, efficiency and robustness of large language models (LLMs). Specifically, we propose meta-buffer to store a series of informative high-level thoughts, namely thought-template, distilled from the problem-solving processes across various tasks. Then for each problem, we retrieve a relevant thought-template and adaptively instantiate it with specific reasoning structures to conduct efficient reasoning. To guarantee the scalability and stability, we further propose buffer-manager to dynamically update the meta-buffer, thus enhancing the capacity of meta-buffer as more tasks are solved. We conduct extensive experiments on 10 challenging reasoning-intensive tasks, and achieve significant performance improvements over previous SOTA methods: 11% on Game of 24, 20% on Geometric Shapes and 51% on Checkmate-in-One. Further analysis demonstrate the superior generalization ability and model robustness of our BoT, while requiring only 12% of the cost of multi-query prompting methods (e.g., tree/graph of thoughts) on average. Notably, we find that our Llama3-8B + BoT has the potential to surpass Llama3-70B model.