# Teacher Supervision over Representation Equivalence Classes

arXiv: https://arxiv.org/abs/2607.03572 · HTML: https://mlsanity.com/paper/2607.03572/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in General knowledge distillation methods, one of 12 directions in Knowledge distillation.
- Builds on 17 of its 23 references that are in the map.
- First posted on arXiv 3 Jul 2026.

## Placed in

- Knowledge distillation / [General knowledge distillation methods](https://mlsanity.com/distill/general-knowledge-distillation-methods/)

## Builds on (most cited first, 17 of 17)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [FitNets: Hints for Thin Deep Nets](https://mlsanity.com/paper/1412.6550/) (2014)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [STaR: Bootstrapping Reasoning With Reasoning](https://mlsanity.com/paper/2203.14465/) (2022)
- [Relational Knowledge Distillation](https://mlsanity.com/paper/1904.05068/) (2019)
- [Similarity-Preserving Knowledge Distillation](https://mlsanity.com/paper/1907.09682/) (2019)
- [Reinforced Self-Training (ReST) for Language Modeling](https://mlsanity.com/paper/2308.08998/) (2023)
- [Git Re-Basin: Merging Models modulo Permutation Symmetries](https://mlsanity.com/paper/2209.04836/) (2022)
- [The Role of Permutation Invariance in Linear Mode Connectivity of Neural Networks](https://mlsanity.com/paper/2110.06296/) (2021)
- [Understanding image representations by measuring their equivariance and equivalence](https://mlsanity.com/paper/1411.5908/) (2014)
- [Revisiting Model Stitching to Compare Neural Representations](https://mlsanity.com/paper/2106.07682/) (2021)
- [Similarity of Neural Network Models: A Survey of Functional and Representational Measures](https://mlsanity.com/paper/2305.06329/) (2023)
- [AgentArk: Distilling Multi-Agent Intelligence into a Single LLM Agent](https://mlsanity.com/paper/2602.03955/) (2026)
- [ProST: Progressive Sub-task Training for Pareto-Optimal Multi-agent Systems Using Small Language Models](https://mlsanity.com/paper/2509.04508/) (2025)

## Abstract (arXiv)

Knowledge distillation is usually framed as a choice of what to match in the teacher - its logits, hidden features, or sample relations - which presupposes that the teacher's representation has absolute coordinates to match. It does not: a pretrained representation is identifiable only up to an orthogonal-and-isotropic-scaling equivalence class, so a student should learn the teacher's equivalence class, not its features. The organizing fact is that capability is the teacher's output function, a class invariant that factors through the quotient by the class action, so an objective recovers capability exactly when it is defined there. This makes absolute feature matching ill-posed, and admissible supervision a matter of targeting class invariants (Gram structure, CKA, principal subspaces) or aligning coordinates first, unifying feature matching, relational distillation, alignment, and grafting in one geometric account. We validate our framework on Qwen2.5 and Llama-3.1. A restoration study recovers a corrupted model's representation (CKA ~ 0.99) but not its capability, and an ablation isolates the cause: output-function (logit) matching drives capability, while matching hidden representations aligns geometry without restoring function. Recovery is confined to the corpus-covered region, and a graft study confirms that boundary overlap predicts transplant success but is necessary, not sufficient.