Large language models are commonly aligned through supervised fine-tuning, yet little is known about how their internal representations evolve during this process. We study alignment dynamics using persistent homology by tracking the topology of activation spaces throughout fine-tuning. Across four transformer language models ranging from 1B to 7B parameters and three alignment objectives corresponding to helpful, harmless, and mixed training data, we find that the majority of topological reorganization occurs during the earliest stages of training. A dense checkpoint analysis reveals a transient peak in topological activity followed by rapid stabilization. We further show that different alignment objectives induce distinguishable topological trajectories, while instruction-tuned and pretrained models exhibit qualitatively different patterns of evolution. Our results suggest that persistent homology provides a complementary perspective on alignment, revealing representation-level changes that are not apparent from behavioral metrics alone.
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.
Preference alignment has substantially improved the observable behavior of large language models, yet it remains unclear what alignment changes internally. Aligned systems still fail under jailbreaks, prompt injection, and retrieval-time corruption, suggesting behavior-level evaluation alone is incomplete. Post-training should leave measurable traces in internal computation. We ask: when an instruction-tuned (IT) model becomes a preference-aligned (PA) model, what geometric structure changes, where do those changes concentrate, and how selectively do they vary across concepts, prompts, and model families? We introduce MENTIS, a geometry-first framework for measuring alignment-induced internal reorganization in paired checkpoints. MENTIS compares IT and PA models using a primary layerwise covariance-based torsion norm (T1), a secondary spectral torsion diagnostic (T2), and an Energy-Radiance-Activation measure (ERA) for depth localization. Across four 7-8B model pairs on LITMUS, our study reveals that alignment-induced change is selective rather than uniform: normative concepts exhibit larger torsion shifts than factual concepts on average; torsion is negatively correlated with contextual entropy; and peak effects localize to architecture-specific mid-to-late layers. The same pattern appears across word-level, prompt-level, and model-level analyses. These results suggest preference alignment leaves structured, depth-localized geometric signatures in internal computation beyond what behavior-level evaluation alone can reveal.
Large language models are now adapted through chains of post-training stages rather than through a single instruction-tuning pass. This paper studies whether such sequential post-training gradually compresses internal representations into low-rank, anisotropic, and homogeneous feature spaces. We define a measurement suite for hidden states, logits, token trajectories, and LoRA updates, and we use it to analyze supervised fine-tuning, preference optimization, safety/refusal tuning, math and code specialization, and long chain-of-thought tuning under controlled stage orderings. The central hypothesis is that excessive representation concentration is not merely a geometric curiosity: it predicts reduced plasticity during later adaptation, weaker out-of-domain generalization, and poorer calibration. We further evaluate lightweight interventions, including mixed-domain replay, feature refresh, representation diversity regularization, and LoRA update decorrelation, as ways to preserve future learnability without giving up the behavioral gains of post-training.