# TIPS: Topological Ill-Posedness Probing and Steering in Large Language Models

arXiv: https://arxiv.org/abs/2606.23590 · HTML: https://mlsanity.com/paper/2606.23590/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 38 references that are in the map.
- First posted on arXiv 22 Jun 2026.

## Builds on (most cited first, 25 of 25)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Representation Engineering: A Top-Down Approach to AI Transparency](https://mlsanity.com/paper/2310.01405/) (2023)
- [Inference-Time Intervention: Eliciting Truthful Answers from a Language Model](https://mlsanity.com/paper/2306.03341/) (2023)
- [Steering Llama 2 via Contrastive Activation Addition](https://mlsanity.com/paper/2312.06681/) (2023)
- [Refusal in Language Models Is Mediated by a Single Direction](https://mlsanity.com/paper/2406.11717/) (2024)
- [AmbigQA: Answering Ambiguous Open-domain Questions](https://mlsanity.com/paper/2004.10645/) (2020)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [ASQA: Factoid Questions Meet Long-Form Answers](https://mlsanity.com/paper/2204.06092/) (2022)
- [SituatedQA: Incorporating Extra-Linguistic Contexts into QA](https://mlsanity.com/paper/2109.06157/) (2021)
- [Asking Clarifying Questions in Open-Domain Information-Seeking Conversations](https://mlsanity.com/paper/1907.06554/) (2019)
- [CLAM: Selective Clarification for Ambiguous Questions with Generative Language Models](https://mlsanity.com/paper/2212.07769/) (2022)
- [Selectively Answering Ambiguous Questions](https://mlsanity.com/paper/2305.14613/) (2023)
- [Aligning Language Models to Explicitly Handle Ambiguity](https://mlsanity.com/paper/2404.11972/) (2024)
- [CLAMBER: A Benchmark of Identifying and Clarifying Ambiguous Information Needs in Large Language Models](https://mlsanity.com/paper/2405.12063/) (2024)
- [Locate, Steer, and Improve: A Practical Survey of Actionable Mechanistic Interpretability in Large Language Models](https://mlsanity.com/paper/2601.14004/) (2026)
- [Asking Clarification Questions to Handle Ambiguity in Open-Domain QA](https://mlsanity.com/paper/2305.13808/) (2023)
- [Large Language Models Struggle with Unreasonability in Math Problems](https://mlsanity.com/paper/2403.19346/) (2024)
- [Detecting (Un)answerability in Large Language Models with Linear Directions](https://mlsanity.com/paper/2509.22449/) (2025)
- [Persistent Topological Features in Large Language Models](https://mlsanity.com/paper/2410.11042/) (2024)
- [ReliableMath: Benchmark of Reliable Mathematical Reasoning on Large Language Models](https://mlsanity.com/paper/2507.03133/) (2025)
- [Sparse Neurons Carry Strong Signals of Question Ambiguity in LLMs](https://mlsanity.com/paper/2509.13664/) (2025)
- [The Shape of Word Embeddings: Quantifying Non-Isometry With Topological Data Analysis](https://mlsanity.com/paper/2404.00500/) (2024)
- [MINT: Minimal Information Neuro-Symbolic Tree for Objective-Driven Knowledge-Gap Reasoning and Active Elicitation](https://mlsanity.com/paper/2602.05048/) (2026)
- [Local Topology Measures of Contextual Language Model Latent Spaces With Applications to Dialogue Term Extraction](https://mlsanity.com/paper/2408.03706/) (2024)

## Abstract (arXiv)

Ill-posed questions, including those involving ambiguity, under-specification, or conflicting statements, may admit no valid answer or multiple plausible answers, posing a significant challenge for large language models (LLMs) despite their otherwise strong performance. Existing work often treats LLM reasoning as a black box and focuses on input-output analysis. Can a compact, unified topological representation of internal model states capture diverse sources of ill-posedness and steer reasoning toward responses appropriate to each source? To this end, we study the internal state from a topological perspective by treating the contextual hidden states of its prompt tokens at a single transformer layer as a point cloud, capturing the input problem's relational structure as encoded by the model. Our analysis shows that zero- and one-dimensional persistent homology---tracking how connected components merge as the distance threshold increases (H0) and the emergence and filling of one-dimensional loops in the filtration (H1), respectively---yield seven compact descriptors of ill-posedness that also serve as control signals for steering model reasoning. Thus, we develop a topology-conditioned steering mechanism. It retrieves topologically similar positive and negative examples for each query to form a local activation-space contrast. The resulting query-specific direction modifies internal states during prefill and decoding, guiding the model's responses toward suitable abstention or clarification tailored to the underlying source of ill-posedness. Empirical evaluations on two datasets and open-weight models spanning four families demonstrate the effectiveness of this compact representation for ill-posedness classification, while topology-conditioning significantly improved steering performance.