# Learning Transfers: Kan Extensions of Topological Invariants of Learned Representations

arXiv: https://arxiv.org/abs/2606.07627 · HTML: https://mlsanity.com/paper/2606.07627/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 45 references that are in the map.
- First posted on arXiv 30 May 2026.

## Builds on (most cited first, 26 of 26)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology](https://mlsanity.com/paper/1812.09764/) (2018)
- [Representation Topology Divergence: A Method for Comparing Neural Network Representations](https://mlsanity.com/paper/2201.00058/) (2021)
- [A Topological Regularizer for Classifiers via Persistent Homology](https://mlsanity.com/paper/1806.10714/) (2018)
- [Connectivity-Optimized Representation Learning via Persistent Homology](https://mlsanity.com/paper/1906.09003/) (2019)
- [Geometry Helps to Compare Persistence Diagrams](https://mlsanity.com/paper/1606.03357/) (2016)
- [Topological Data Analysis of Decision Boundaries with Application to Model Selection](https://mlsanity.com/paper/1805.09949/) (2018)
- [Metrics for generalized persistence modules](https://mlsanity.com/paper/1312.3829/) (2013)
- [Computing the interleaving distance is NP-hard](https://mlsanity.com/paper/1811.09165/) (2018)
- [Persistent Homology as Stopping-Criterion for Voronoi Interpolation](https://mlsanity.com/paper/1911.02922/) (2019)
- [Estimate of the Neural Network Dimension using Algebraic Topology and Lie Theory](https://mlsanity.com/paper/2004.02881/) (2020)
- [Kan Extensions in Data Science and Machine Learning](https://mlsanity.com/paper/2203.09018/) (2022)
- [Kan Extension Transformers: A Categorical Unification of Attention, Diffusion, and Predict-Detach Self-Conditioning](https://mlsanity.com/paper/2605.27259/) (2026)
- [Domain Adaptation by Topology Regularization](https://mlsanity.com/paper/2101.12102/) (2021)
- [Fast Left Kan Extensions Using The Chase](https://mlsanity.com/paper/2205.02425/) (2022)
- [Topological Signatures of ReLU Neural Network Activation Patterns](https://mlsanity.com/paper/2510.12700/) (2025)
- [Homological Time Series Analysis of Sensor Signals from Power Plants](https://mlsanity.com/paper/2106.02493/) (2021)
- [Learning Is a Kan Extension](https://mlsanity.com/paper/2502.13810/) (2025)
- [Kan Approximations of the Persistent Homology Transform](https://mlsanity.com/paper/2507.22816/) (2025)

## Abstract (arXiv)

When a learned representation is reused for a new task, which of its structure should survive? A task is a small category, a change of task is a functor J, and the invariants of a trained representation form a functor F. The left Kan extension operatornameLanJF is the invariant that the source and the change of task force on the target: by its universal property, every comparison of the source with an observed target invariant factors through it. The transfer discrepancy is the largest distance, over the target objects, between this extension and the observed invariant. For persistent homology the extension is a cokernel over comma categories, and a functorial version of the discrepancy, which also compares the maps between target objects, is 1-Lipschitz under interleavings. On finite samples the discrepancy is a bottleneck distance, computed without approximation. For data glued from two pieces whose exclusive parts lie farther apart than the filtration scale, a Mayer-Vietoris sequence identifies what extending homology loses, while extending chains loses nothing. For a merge of labelled point clouds, the degree-zero discrepancy vanishes if and only if some 1-Lipschitz classifier has a margin above the filtration scale at every sample point. In experiments on sampled manifolds, trained encoders and pretrained image models, every exact prediction holds on every sample. The discrepancy of truncated invariants recovers a hidden merge of separated parts far above chance and tells the intended target of a merge from a permuted control, but across pretrained models it does not track the accuracy of a linear probe.