# Topology-Enhanced Alignment for Large Language Models: Trajectory Topology Loss and Topological Preference Optimization

arXiv: https://arxiv.org/abs/2605.07172 · HTML: https://mlsanity.com/paper/2605.07172/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in LLM preference optimization methods, one of 10 directions in LLM alignment / RLHF / preference optimisation.
- Builds on 19 of its 32 references that are in the map.
- First posted on arXiv 8 May 2026.

## Placed in

- LLM alignment / RLHF / preference optimisation / [LLM preference optimization methods](https://mlsanity.com/alignment/llm-preference-optimization-methods/)

## Builds on (most cited first, 19 of 19)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks](https://mlsanity.com/paper/1908.10084/) (2019)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Learning to summarize from human feedback](https://mlsanity.com/paper/2009.01325/) (2020)
- [A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks](https://mlsanity.com/paper/1610.02136/) (2016)
- [A Simple Unified Framework for Detecting Out-of-Distribution Samples and Adversarial Attacks](https://mlsanity.com/paper/1807.03888/) (2018)
- [RealToxicityPrompts: Evaluating Neural Toxic Degeneration in Language Models](https://mlsanity.com/paper/2009.11462/) (2020)
- [Enhancing Chat Language Models by Scaling High-quality Instructional Conversations](https://mlsanity.com/paper/2305.14233/) (2023)
- [How Contextual are Contextualized Word Representations? Comparing the Geometry of BERT, ELMo, and GPT-2 Embeddings](https://mlsanity.com/paper/1909.00512/) (2019)
- [Network Dissection: Quantifying Interpretability of Deep Visual Representations](https://mlsanity.com/paper/1704.05796/) (2017)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Emergence of Invariance and Disentanglement in Deep Representations](https://mlsanity.com/paper/1706.01350/) (2017)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology](https://mlsanity.com/paper/1812.09764/) (2018)
- [Neural Anisotropy Directions](https://mlsanity.com/paper/2006.09717/) (2020)
- [Topological Data Analysis for Neural Network Analysis: A Comprehensive Survey](https://mlsanity.com/paper/2312.05840/) (2023)

## Abstract (arXiv)

Alignment of large language models (LLMs) via SFT and RLHF/DPO typically ignores the global geometry of the representation space, relying instead on local token likelihoods or scalar scores. We view generation as tracing a semantic trajectory in hidden space and propose a topology-enhanced alignment framework that regularizes these trajectories using 0-dimensional persistent homology. First, for SFT, we introduce Trajectory Topology Loss (TTL). Treating prompt and gold-answer embeddings as a mixed point cloud, we use a 0D persistent homology algorithm to extract "prompt-answer bridges." TTL aligns the model's actual update direction with these topological bridges rather than arbitrary directions. Second, for DPO, we propose Topological Preference Optimization (TPO). TPO constructs topic-specific semantic preference vectors and aligns the improvement direction between rejected and chosen responses with these vectors in an intermediate hidden layer. We also introduce a dynamic weighting scheme to balance DPO and TPO losses. Evaluating on Qwen2.5-7B-Instruct using UltraChat and Anthropic HH-RLHF, our topology-enhanced objectives consistently outperform strong non-topological baselines (e.g., per-example, nearest-neighbor, random regularizers) on automatic preference metrics and LLM-judge evaluations, while maintaining or improving toxicity. Results show persistent homology and trajectory geometry offer a promising direction for controllable alignment.