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Learning Global Hypothesis Space for Enhancing Synergistic Reasoning Chain

Jiaquan Zhang, Chaoning Zhang, Shuxu Chen, Xudong Wang, Zhenzhen Huang, Pengcheng Zheng, Shuai Yuan, Sheng Zheng, Qigan Sun, Jie Zou, Lik-Hang Lee, Yang Yang

arXiv:2602.09794v2cs.AI

TL;DR

Existing CoT approaches are vulnerable to cascading early errors and lack mechanisms for globally coordinating hypotheses or structurally filtering reasoning. GHS-TDA builds a global hypothesis graph and analyzes it with persistent homology, achieving consistent gains over strong baselines across multiple reasoning benchmarks.

  • Problem

    Existing CoT methods rely on autoregressive generation and lack global coordination and structured analysis for controlling errors, redundancy, and salient reasoning features.

  • Method

    GHS-TDA integrates diverse candidate paths in a Global Hypothesis Graph and applies persistent homology with Betti stability checks to extract robust H0 backbones and H1 loops.

  • Results

    GHS-TDA consistently outperforms existing methods in accuracy, consistency, and interpretability across benchmarks including GSM8K, MATH, OlympiadBench, HotpotQA, MuSiQue, BBH, and LongBench.

  • Takeaways & Limitations

    Combining structural integration with topological robustness provides a principled foundation for more reliable and transparent reasoning systems.

Abstract

from arXiv · show

Chain-of-Thought (CoT) has been shown to significantly improve the reasoning accuracy of large language models (LLMs) on complex tasks. However, due to the autoregressive, step-by-step generation paradigm, existing CoT methods suffer from two fundamental limitations. First, the reasoning process is highly sensitive to early decisions: once an initial error is introduced, it tends to propagate and amplify through subsequent steps, while the lack of a global coordination and revision mechanism makes such errors difficult to correct, ultimately leading to distorted reasoning chains. Second, current CoT approaches lack structured analysis techniques for filtering redundant reasoning and extracting key reasoning features, resulting in unstable reasoning processes and limited interpretability. To address these issues, we propose GHS-TDA. GHS-TDA first constructs a semantically enriched global hypothesis graph to aggregate, align, and coordinate multiple candidate reasoning paths, thereby providing alternative global correction routes when local reasoning fails. It then applies topological data analysis based on persistent homology to capture stable multi-scale structures, remove redundancy and inconsistencies, and extract a more reliable reasoning skeleton. By jointly leveraging reasoning diversity and topological stability, GHS-TDA achieves self-adaptive convergence, produces high-confidence and interpretable reasoning paths, and consistently outperforms strong baselines in terms of both accuracy and robustness across multiple reasoning benchmarks.

1 INTRODUCTION

Existing CoT methods improve reasoning by decomposing problems into steps but remain vulnerable to cascading early errors, fragmented hypotheses, redundancy, and limited structural control. GHS-TDA addresses these issues by constructing a global hypothesis graph and applying persistent-homology analysis, with experiments showing consistent gains across multiple benchmarks.

  • CoT improves reasoning accuracy and interpretability by decomposing complex problems into coherent intermediate steps.
  • Autoregressive CoT generation makes reasoning susceptible to early errors that propagate through later steps and lacks structured control over redundancy and salient features.
  • Persistent homology extracts stable reasoning backbones and self-consistent loops from the global hypothesis graph for more interpretable reasoning chains.
  • GHS-TDA constructs a global hypothesis graph that integrates diverse reasoning paths through multi-role interactions, unification, conflict detection, and closure inference.
  • GHS-TDA is evaluated on GSM8K, MATH, OlympiadBench, HotpotQA, MuSiQue, BBH, and LongBench, consistently outperforming existing methods in accuracy, consistency, and interpretability.

2 RELATED WORK

Prior work uses increasingly structured prompting to improve LLM reasoning, but existing methods still lack global hypothesis integration and systematic analysis of reasoning-chain structure. TDA, particularly persistent homology, offers a framework for identifying stable topological structures in reasoning chains.

  • Chain-of-Thought prompting improves LLM reasoning accuracy and interpretability by decomposing complex problems into explicit intermediate steps.
  • Tree-of-Thought, Graph-of-Thought, and Atom-of-Thought extend reasoning into tree, graph, and atomic structures, enabling parallel branch exploration and evidence reuse.
  • Existing approaches rely mainly on local heuristics, lacking global integration of diverse hypotheses and systematic analysis of connectivity, consistency, and redundancy.
  • Persistent homology analyzes how connected components and loops evolve across scales, extracting structural features stable under noise and local perturbations.
  • Applying TDA to LLM reasoning chains can identify persistent connected components and self-consistent loops within their graph-structured semantic and logical organization.

3 METHOD

GHS-TDA constructs a unified Global Hypothesis Graph from multiple reasoning paths, then applies feature-based topological analysis to extract stable backbones and self-consistent loops for reasoning and answer aggregation.

  • Global Hypothesis Space Modeling: GHS-TDA samples multiple reasoning paths and merges semantically equivalent hypotheses into a Global Hypothesis Graph that preserves diversity while eliminating redundancy.The graph retains alternative hypotheses and their semantic and logical dependencies in a shared reasoning space.
  • Global Hypothesis Space Modeling: Graph nodes store text, canonicalized forms, confidence, and progress, while edges encode adjacency, support, refutation, and other semantic-logical dependencies.Canonicalized forms support equivalence testing, and merged nodes inherit incident edges while aggregating confidence and progress information.
  • Skeleton Extraction: Each node is embedded with semantic, graph-structural, and uncertainty features, whose normalized combination defines the space used for topological analysis.Uncertainty is represented as uv = −log(confidencev + 10^-6), while graph features include progress, positional encoding, and centrality.
  • Skeleton Extraction: A sparsified KNN graph supports Vietoris–Rips filtration, and persistent homology selects long-lived H0 clusters and H1 loops as significant reasoning structures.The filtration preserves salient topological features while reducing complexity; feature significance is based on lifespan.
  • Skeleton Extraction: Skeleton construction maps clusters and loops back to the graph, builds shortest-path backbones, and can reroute them through principal loops for verification.The resulting skeleton supports confidence/persistence-weighted answer voting and reports contributing paths, edge weights, and loop lifespan.
  • Implementation: The implementation uses fixed embedding, sampling, filtration, and call-budget settings, including GUDHI persistence computation up to H1.The reported settings include five random seeds, temperature 0.7, top-p 0.95, and at most 16 LLM calls per example.

4 EXPERIMENT

GHS-TDA is evaluated across diverse models, benchmarks, baselines, accuracy measures, interpretability criteria, robustness tests, predictive analyses, and efficiency metrics. Results show strong accuracy, compact and interpretable path selection, robustness to perturbations, predictive value of H1 persistence, and reduced inference cost.

  • Experimental setup: Experiments use three backbone models, eight reasoning benchmarks, nine representative baselines, and exact-match accuracy as the primary metric.The evaluation covers GPT-4o-mini, Qwen-Turbo, and DeepSeek-V3 across arithmetic, mathematics, multi-hop, and long-context reasoning tasks.
  • Main results: GHS-TDA achieves the highest or near-highest results across datasets and backbones, with average EM scores of 68.0%, 67.6%, and 68.3%.These averages on GPT-4o-mini, Qwen-Turbo, and DeepSeek-V3 exceed AoT's corresponding 66.9%, 66.8%, and 67.3%.
  • Path selection analysis: The TDA Skeleton reaches 83.9% accuracy with 8.7 steps and low confidence variance, while closely matching human interpretability ratings.Its human ratings are 4.4 for clarity, 4.5 for coherence, 4.7 for credibility, and 4.3 for conciseness.
  • Robustness analysis: Under adversarial paraphrase perturbations, GHS-TDA's selected path loses 2.4 points in accuracy and changes answers at a 2.9% rate, versus 7.4% for Max-Confidence.The results support stronger robustness for paths selected by topological stability than for locally confidence-based selection.
  • Predictive persistence analysis: H1 persistence is positively associated with correctness, with a global Spearman correlation of 0.349, roughly 3.5-times higher correctness odds per standard deviation, and AUC 0.74.Per-dataset AUC values remain within 0.70–0.78 across all eight benchmarks, with HotpotQA reaching 0.778.
  • Efficiency analysis: GHS-TDA reduces token consumption by about 26.8% versus ToT and 35.7% versus AoT, while maintaining a fixed upper bound of 19 LLM calls.The call-count and token results indicate lower redundant generation and inference cost than the compared multi-path methods.

5 CONCLUSION

GHS-TDA integrates global hypothesis graph construction with topological data analysis to produce robust reasoning. Across multiple benchmarks, it improves accuracy and interpretability, while topological persistence functions as a task-agnostic reliability indicator.

  • GHS-TDA unifies diverse reasoning paths into a coherent hypothesis space and extracts stable backbones and self-consistent loops via persistent homology.
  • Experiments across multiple benchmarks demonstrate consistent gains over strong baselines in reasoning accuracy and interpretability.
  • Topological persistence serves as a task-agnostic indicator of reasoning reliability.

A.2 DEFINITION OF SUPPORT AND REFUTATION

The method reduces the cost of incorporating logical relations by evaluating a restricted candidate set of graph pairs. An LLM labels these pairs as SUPPORT, REFUTE, or NEUTRAL, and the resulting relation code modifies the TDA distance.

  • The candidate set combines longitudinal derivation edges with lateral unconnected pairs that may represent competing hypotheses.Longitudinal pairs encode potential premise-to-conclusion relations, while lateral pairs capture same-stage alternatives.
  • The reduced candidate set has size K ≪ |V|^2, avoiding evaluation of all node pairs.
  • Each candidate pair receives one of three logical labels: SUPPORT, REFUTE, or NEUTRAL.Pairs are processed in chunks, with one LLM call per chunk.
  • The inferred relation code R(v_i, v_j) is added to the TDA distance alongside embedding, graph, and uncertainty terms.The distance is defined as d(v_i, v_j) = α(1 −⟨e_i, e_j⟩) + β∥ϕ_graph(i) −ϕ_graph(j)∥_1 + ν(u_i + u_j) + δ·R(v_i, v_j).
  • REFUTE assigns a large positive penalty, SUPPORT assigns a negative attraction, and NEUTRAL leaves the original distance unchanged.REFUTE pushes nodes apart, whereas SUPPORT draws them closer to encourage coherent connections.

A.3 PERSISTENCE–ACCURACY ANALYSIS ACROSS DATASETS

Across eight datasets, higher H1 persistence is consistently associated with higher reasoning accuracy, with sharp gains at low-to-moderate persistence and saturation at higher values. The relationship varies by task type, while OlympiadBench shows a lower plateau indicating a boundary for first-order topological features.

  • Across all eight datasets, reasoning accuracy increases monotonically with H1 persistence.The analysis uses logistic fits with binned means overlaid.
  • Accuracy gains are sharp for H1 ∈ [0, 2] and gradually saturate at higher persistence values.The saturation reflects a ceiling once loop stability is sufficient.
  • Arithmetic and short-chain reasoning: On GSM8K, accuracy rises steeply and quickly saturates near perfect accuracy, while MATH shows a smoother monotonic increase.Persistent loops are interpreted as self-verification mechanisms for arithmetic tasks, whereas complex derivations require higher persistence.
  • Long-context and multi-hop reasoning: HotpotQA, LongBench, and MuSiQue show steep low-persistence gains and earlier saturation, highlighting stable loops for multi-evidence integration.HotpotQA reaches near-perfect accuracy once persistence exceeds moderate values.
  • Knowledge-intensive and challenging reasoning: MMLU-CF shows steadily increasing accuracy with persistence, whereas OlympiadBench improves but reaches a slightly lower plateau.The lower OlympiadBench plateau suggests that creative steps or lengthy derivations may not be fully captured by first-order topological features.
  • The analysis recommends persistence-aware path selection using thresholding such as H1 ≥ 1 or weighted scoring.

A.4 STABILITY ANALYSIS

The GHS-TDA pipeline remains computationally manageable on sparse graphs and its topological signatures are stable under embedding perturbations. Runtime is dominated by embedding and KNN construction, while dominant H1 features persist under tested noise levels.

  • GHS-TDA processes roughly 80–120 nodes per GSM8K or MATH instance with KNN neighborhood size fixed at 15.
  • End-to-end graph construction takes 25–60 ms per instance on an RTX 4090 system, with embedding and KNN construction dominating runtime.
  • Vietoris–Rips filtration uses 150–400 MB for graphs of around 100 nodes and remains below 1 GB at approximately 200 nodes.The filtration is computed up to the first homology group on sparse graphs.
  • Under Gaussian embedding noise from 0 to 0.1, bottleneck distances range from 0.01 to 0.10 and remain below dominant feature lifetimes of 0.3 to 0.8.

A.5 GENERALIZATION

GHS-TDA generalizes across base models and benchmarks, maintaining consistent performance gains without additional supervision or external knowledge. The reported results attribute this robustness to the reasoning mechanism rather than model size alone.

  • On Llama 3-8B, GHS-TDA achieves an average score of 63.88, surpassing CoT, ToT, and AoT by 6.9, 3.0, and 1.1 points.
  • On Qwen2-14B, GHS-TDA reaches 62.35, exceeding the corresponding baselines by 6.75, 2.7, and 1.5 points.
  • The improvements occur without additional supervision or external knowledge, indicating that the gains come from the reasoning mechanism rather than model size.
  • Additional experiments report that GHG-TDA achieves average scores of 72.4 on GPT-4o and 72.8 on Claude 3.5 Sonnet, outperforming AoT, GoT, ToT, and CoT on all benchmarks.

A.6 CORRELATION ANALYSIS

The paper analyzes how H1 persistence relates to reasoning correctness across datasets and separately for deepseek-V3. The analysis uses correlation and discrimination measures to quantify this relationship.

  • Tables 10 and 11 quantify how H1 persistence relates to successful problem-solving across diverse reasoning tasks and for deepseek-V3.
  • Table 9 presents performance comparisons across two base models, providing a separate cross-model evaluation context for the reasoning methods.
  • Table 10 is described as reporting correlation between H1 persistence and reasoning correctness across datasets.

A.7 PARAMETER SETTINGS AND TUNING STRATEGIES

The framework tunes graph compression and topological sensitivity through distance weights, node merging, feature selection, and loop thresholds. The reported analyses connect these choices to redundancy control, connectivity, robustness, and interpretability.

  • Hybrid distance: Increase α for precise reasoning, β for long dependencies, and γ for noisy outputs.
  • Topological features: K = 5 is the default feature budget, with a practical range of 3–8; too few features omit plausible chains, whereas too many introduce noisy cycles.
  • Parameter overview: The main hyperparameters are θmerge, distance weights (α, β, γ), the topological feature budget K, and loop threshold δ.
  • Hybrid distance: Hybrid distance uses α + β + γ = 1 with default weights (0.6, 0.3, 0.1), making semantic similarity dominant while structure supports long dependencies and uncertainty regularizes noise.
  • Robustness: Persistent homology filters short-lived features caused by imperfect confidence estimates, leaving outcomes driven primarily by persistence and structural consistency.
  • Node merging: The default θmerge = 0.85 balances redundancy removal and connectivity; thresholds around 0.70 can cause semantic collapse, while approximately 0.80–0.90 is more effective.
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