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The Topology of Ill-Posed Questions: Persistent Hom... | AI Research

Key Takeaways

  • The Topology of Ill-Posed Questions: Persistent Homology for Detection and Steering in LLMs Large language models often struggle when faced with "ill-posed"...
  • 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.
Paper AbstractExpand

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.

The Topology of Ill-Posed Questions: Persistent Homology for Detection and Steering in LLMs
Large language models often struggle when faced with "ill-posed" questions—queries that are ambiguous, contradictory, or missing necessary context. While existing research typically analyzes these issues by looking at the final text output, this paper explores whether the internal "thought process" of an LLM can reveal the nature of these questions. By treating the model’s internal hidden states as a geometric shape, the authors demonstrate that they can detect ill-posedness more accurately and steer the model toward more helpful, clarified responses.

Mapping the Geometry of Thought

The researchers propose that when an LLM processes a prompt, its internal hidden states for each token form a "point cloud" within the model's layers. To understand the structure of this cloud, they use a technique from topological data analysis called persistent homology. This method tracks how different groups of tokens connect to one another as the model processes information. By focusing on zero-dimensional persistent homology, the authors measure how and when these token groups merge, providing a mathematical signature of the question's complexity.

Compact Topological Descriptors

To make this complex geometric data usable, the authors summarize each layer of the model using three specific metrics: the average scale at which token groups merge, the distribution of these merge events, and the dominance of the largest connections. By combining these three descriptors across every layer of the transformer, they create a "topology representation" of the question. This compact vector acts as a fingerprint that captures the underlying uncertainty or ambiguity of the query, regardless of the specific words used.

Improving Detection and Response

This topological approach significantly outperforms traditional methods for identifying ill-posed questions. Across three different open-weight LLMs, the topology-based features improved classification accuracy on benchmarks like AmbigQA and SituatedQA. Beyond just detecting problems, the authors introduced "topology-conditioned activation steering." This technique identifies similar examples from the model's history and uses them to create a custom intervention. Instead of forcing the model to use a generic refusal, this method helps the LLM provide a response that is specifically tailored to the source of the confusion, such as asking for clarification or acknowledging missing information, while still preserving the original intent of the user's question.

Key Takeaways

The study demonstrates that the internal geometry of an LLM’s hidden states is a powerful, interpretable signal for understanding why a model might fail. By moving away from black-box output analysis and toward a structural, layer-wise view of internal activations, the authors provide a more nuanced way to handle complex or poorly formed user queries. These findings suggest that topological analysis is not only a diagnostic tool but also a viable mechanism for guiding LLMs toward more reliable and grounded behavior.

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