# Embedding Trajectory for Out-of-Distribution Detection in Mathematical Reasoning

arXiv: https://arxiv.org/abs/2405.14039 · HTML: https://mlsanity.com/paper/2405.14039/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 44 of its 62 references that are in the map.
- First posted on arXiv 22 May 2024.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 30 of 44)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [BERTScore: Evaluating Text Generation with BERT](https://mlsanity.com/paper/1904.09675/) (2019)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks](https://mlsanity.com/paper/1610.02136/) (2016)
- [SimCSE: Simple Contrastive Learning of Sentence Embeddings](https://mlsanity.com/paper/2104.08821/) (2021)
- [A Simple Unified Framework for Detecting Out-of-Distribution Samples and Adversarial Attacks](https://mlsanity.com/paper/1807.03888/) (2018)
- [Enhancing The Reliability of Out-of-distribution Image Detection in Neural Networks](https://mlsanity.com/paper/1706.02690/) (2017)
- [Energy-based Out-of-distribution Detection](https://mlsanity.com/paper/2010.03759/) (2020)
- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [Are NLP Models really able to Solve Simple Math Word Problems?](https://mlsanity.com/paper/2103.07191/) (2021)
- [Generalized Out-of-Distribution Detection: A Survey](https://mlsanity.com/paper/2110.11334/) (2021)
- [Automatic Chain of Thought Prompting in Large Language Models](https://mlsanity.com/paper/2210.03493/) (2022)
- [Out-of-Distribution Detection with Deep Nearest Neighbors](https://mlsanity.com/paper/2204.06507/) (2022)
- [Trustworthy LLMs: a Survey and Guideline for Evaluating Large Language Models' Alignment](https://mlsanity.com/paper/2308.05374/) (2023)
- [Solving General Arithmetic Word Problems](https://mlsanity.com/paper/1608.01413/) (2016)
- [On the Importance of Gradients for Detecting Distributional Shifts in the Wild](https://mlsanity.com/paper/2110.00218/) (2021)
- [TrustLLM: Trustworthiness in Large Language Models](https://mlsanity.com/paper/2401.05561/) (2024)
- [Pitfalls of In-Domain Uncertainty Estimation and Ensembling in Deep Learning](https://mlsanity.com/paper/2002.06470/) (2020)
- [A Simple Fix to Mahalanobis Distance for Improving Near-OOD Detection](https://mlsanity.com/paper/2106.09022/) (2021)
- [Out-of-Distribution Detection and Selective Generation for Conditional Language Models](https://mlsanity.com/paper/2209.15558/) (2022)
- [A Careful Examination of Large Language Model Performance on Grade School Arithmetic](https://mlsanity.com/paper/2405.00332/) (2024)
- [Training Very Deep Networks](https://mlsanity.com/paper/1507.06228/) (2015)

## Built on by (most cited first, 18 of 18)

- [Sampling-Efficient Test-Time Scaling: Self-Estimating the Best-of-N Sampling in Early Decoding](https://mlsanity.com/paper/2503.01422/) (2025)
- [Latent Space Chain-of-Embedding Enables Output-free LLM Self-Evaluation](https://mlsanity.com/paper/2410.13640/) (2024)
- [PolyMath: Evaluating Mathematical Reasoning in Multilingual Contexts](https://mlsanity.com/paper/2504.18428/) (2025)
- [Consistent Paths Lead to Truth: Self-Rewarding Reinforcement Learning for LLM Reasoning](https://mlsanity.com/paper/2506.08745/) (2025)
- [Out-of-Distribution Detection: A Task-Oriented Survey of Recent Advances](https://mlsanity.com/paper/2409.11884/) (2024)
- [The Geometry of Reasoning: Flowing Logics in Representation Space](https://mlsanity.com/paper/2510.09782/) (2025)
- [GEM: Gaussian Embedding Modeling for Out-of-Distribution Detection in GUI Agents](https://mlsanity.com/paper/2505.12842/) (2025)
- [Building Trustworthy AI for Materials Discovery: From Autonomous Laboratories to Z-scores](https://mlsanity.com/paper/2512.01080/) (2025)
- [Amplify Adjacent Token Differences: Enhancing Long Chain-of-Thought Reasoning with Shift-FFN](https://mlsanity.com/paper/2505.17153/) (2025)
- [On the Overscaling Curse of Parallel Thinking: System Efficacy Contradicts Sample Efficiency](https://mlsanity.com/paper/2601.21619/) (2026)
- [AP-OOD: Attention Pooling for Out-of-Distribution Detection](https://mlsanity.com/paper/2602.06031/) (2026)
- [Translationese-index: Using Likelihood Ratios for Graded and Generalizable Measurement of Translationese](https://mlsanity.com/paper/2507.12260/) (2025)
- [CONFIDE: Hallucination Assessment for Reliable Biomolecular Structure Prediction and Design](https://mlsanity.com/paper/2512.02033/) (2025)
- [Useful Features, Backward Scores: OOD in Language-Model Trajectories](https://mlsanity.com/paper/2605.00269/) (2026)
- [Autonomous Drift Learning in Data Streams: A Unified Perspective](https://mlsanity.com/paper/2605.01295/) (2026)
- [Micro-Defects Expose Macro-Fakes: Detecting AI-Generated Images via Local Distributional Shifts](https://mlsanity.com/paper/2605.09296/) (2026)
- [QueST: Persistent Queries as Semantic Monitors for Drift Suppression in Long-Horizon Tracking](https://mlsanity.com/paper/2605.09513/) (2026)
- [Understanding LLM Behavior in Multi-Target Cross-Lingual Summarization](https://mlsanity.com/paper/2606.01252/) (2026)

## Abstract (arXiv)

Real-world data deviating from the independent and identically distributed (i.i.d.) assumption of in-distribution training data poses security threats to deep networks, thus advancing out-of-distribution (OOD) detection algorithms. Detection methods in generative language models (GLMs) mainly focus on uncertainty estimation and embedding distance measurement, with the latter proven to be most effective in traditional linguistic tasks like summarization and translation. However, another complex generative scenario mathematical reasoning poses significant challenges to embedding-based methods due to its high-density feature of output spaces, but this feature causes larger discrepancies in the embedding shift trajectory between different samples in latent spaces. Hence, we propose a trajectory-based method TV score, which uses trajectory volatility for OOD detection in mathematical reasoning. Experiments show that our method outperforms all traditional algorithms on GLMs under mathematical reasoning scenarios and can be extended to more applications with high-density features in output spaces, such as multiple-choice questions.