# Fisher-IRG: Fisher-Induced Local Invariant Representation Geometry across Language and Vision Models

arXiv: https://arxiv.org/abs/2609.36458 · HTML: https://mlsanity.com/paper/2609.36458/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 18 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 14 of 14)

- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [SimCSE: Simple Contrastive Learning of Sentence Embeddings](https://mlsanity.com/paper/2104.08821/) (2021)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [BERT Rediscovers the Classical NLP Pipeline](https://mlsanity.com/paper/1905.05950/) (2019)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [Insights on representational similarity in neural networks with canonical correlation](https://mlsanity.com/paper/1806.05759/) (2018)
- [Understanding image representations by measuring their equivariance and equivalence](https://mlsanity.com/paper/1411.5908/) (2014)
- [Visualizing and Measuring the Geometry of BERT](https://mlsanity.com/paper/1906.02715/) (2019)
- [Emergence of Invariance and Disentanglement in Deep Representations](https://mlsanity.com/paper/1706.01350/) (2017)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Pulling back information geometry](https://mlsanity.com/paper/2106.05367/) (2021)
- [FishBack: Pullback Fisher Geometry for Optimal Activation Steering in Transformers](https://mlsanity.com/paper/2605.17231/) (2026)
- [Invariant Features in Language Models: Geometric Characterization and Model Attribution](https://mlsanity.com/paper/2605.06458/) (2026)

## Abstract (arXiv)

Semantic-preserving transformations can induce substantial motion in learned representations, while small changes may strongly affect model predictions, raising a basic question: what local metric best captures semantically consequential variation? We propose Fisher-induced invariant representation geometry (Fisher-IRG), which measures local representation directions through their predictive sensitivity. Around each representation, we construct semantic-preserving and semantic-changing neighborhoods, aggregate their local Fisher information, and recover invariant directions through a contrastive generalized eigenvalue problem. Controlled displacement analyses first show that comparable Euclidean motion can have substantially different predictive consequences, supporting the need for a predictive geometry. Across language and vision models, Fisher-IRG yields stronger semantic-versus-nuisance predictive selectivity and generally more reproducible subspaces than covariance-based geometry, while recovering systematically distinct local directions. Representation interventions further localize semantic effects to the Fisher-derived subspace, and held-out separation and retrieval show that the recovered geometry generalizes beyond the discovery neighborhoods. These results support Fisher-IRG as a principled framework for characterizing local invariant representation geometry.