The Topology of Ill-Posed Questions: Persistent Homology for Detection and Steering in LLMs

2026-06-22

AI Summary

This work addresses the challenge that large language models struggle to generate effective responses when confronted with ill-posed questions—those that are ambiguous, underspecified, or self-contradictory. The authors introduce persistent homology to this task for the first time, modeling hidden contextual states across transformer layers as point clouds and extracting their geometric and topological features via zero-dimensional persistent homology, including average finite persistence, normalized persistence entropy, and maximal persistence concentration. These features form a cross-layer topological representation used to detect question ill-posedness, which in turn informs a novel topologically conditioned activation intervention mechanism to guide response generation. Experiments demonstrate that the proposed approach achieves ill-posed question classification accuracies of 78.9%, 88.5%, and 69.6% on AmbigQA, SituatedQA, and CLAMBER, respectively, and improves the rate of acceptable answers from 61.4% to 70.6%, with grounded acceptable answers rising from 11.9% to 16.4%.

Abstract

Ill-posed questions, including ambiguous, underspecified, or contradictory queries, may admit no valid answer or multiple plausible answers, posing a challenge for large language models (LLMs). Existing approaches largely analyze ill-posedness through model outputs and often focus on specific subclasses. We investigate whether diverse sources of ill-posedness can be represented within a unified topology of LLM internal states and whether this structure can be used to steer response behavior. We model the contextual hidden states of prompt tokens at each transformer layer as a point cloud and characterize its geometry using finite zero-dimensional persistent homology. Each layer is summarized by three compact descriptors: mean finite lifetime, normalized lifetime entropy, and largest-lifetime concentration. Concatenating these descriptors across layers yields a topology representation of the question. We further introduce topology-conditioned activation steering, which retrieves topologically similar examples and constructs query-specific activation interventions that encourage source-aware clarification or abstention. Across three open-weight LLMs, topology features consistently outperform prompt-based and pooled-hidden-state baselines for ill-posedness classification, improving average accuracy from \(67.4\%\) to \(78.9\%\) on AmbigQA, from \(79.9\%\) to \(88.5\%\) on SituatedQA, and from \(57.6\%\) to \(69.6\%\) on CLAMBER 9-way classification. Topology-conditioned steering increases the average total acceptable response rate from \(61.4\%\) to \(70.6\%\) and grounded acceptable responses from \(11.9\%\) to \(16.4\%\). These results show that persistent homology provides both an interpretable representation of ill-posedness and an effective mechanism for targeted response steering.

Problem

• ill-posed questions
• large language models
• persistent homology
• topology
• response steering

Innovation

• persistent homology
• topological representation
• activation steering
• ill-posed question detection
• LLM internal states