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

Large language models (LLMs) often struggle with ill-posed questions—queries that are ambiguous, underspecified, or contradictory. While previous research has focused on analyzing model outputs or using black-box techniques to detect these issues, this paper asks whether the internal, layer-wise representations of a question can be modeled as a unified topological space to better detect and steer model behavior.

The authors treat the contextual hidden states of prompt tokens at each transformer layer as a point cloud. They apply persistent homology—a tool from topological data analysis—to characterize the connectivity of these token clouds. Specifically, they use zero-dimensional persistent homology (H0) to track how token groups merge as distance thresholds increase. Each layer is summarized by three compact descriptors: mean finite lifetime, normalized lifetime entropy, and the concentration of the largest lifetimes. These descriptors are concatenated across all layers to form a low-dimensional topological representation of the question.

The topological representation consistently outperforms baseline methods (such as prompt-based or pooled-hidden-state approaches) in detecting ill-posedness. Across three open-weight LLM families, the method improved classification accuracy significantly on benchmarks like AmbigQA, SituatedQA, and CLAMBER. Furthermore, the authors introduce a topology-conditioned activation steering mechanism. By retrieving topologically similar examples, the model can perform query-specific interventions that encourage clarification or abstention, rather than relying on generic refusal templates. This approach successfully increased the rate of grounded, acceptable responses.

This work provides a bridge between topological data analysis and LLM interpretability. By showing that ill-posedness is encoded in the geometric structure of internal token states, the authors offer a more granular way to understand how models process uncertainty. This allows for targeted steering that preserves the semantic content of the original query while guiding the model toward more helpful, source-aware responses.