# Certified Topological Interaction in Neural Representations: Exact Tests and the Statistic They Require

arXiv: https://arxiv.org/abs/2609.08561 · HTML: https://mlsanity.com/paper/2609.08561/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 58 references that are in the map.
- First posted on arXiv 8 Sep 2026.

## Builds on (most cited first, 30 of 35)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Fantastic Generalization Measures and Where to Find Them](https://mlsanity.com/paper/1912.02178/) (2019)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [Revisiting Classifier Two-Sample Tests](https://mlsanity.com/paper/1610.06545/) (2016)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology](https://mlsanity.com/paper/1812.09764/) (2018)
- [On the geometry of generalization and memorization in deep neural networks](https://mlsanity.com/paper/2105.14602/) (2021)
- [Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks](https://mlsanity.com/paper/2111.13171/) (2021)
- [A Topological Regularizer for Classifiers via Persistent Homology](https://mlsanity.com/paper/1806.10714/) (2018)
- [Computing the Testing Error without a Testing Set](https://mlsanity.com/paper/2005.00450/) (2020)
- [How deep is deep enough? -- Quantifying class separability in the hidden layers of deep neural networks](https://mlsanity.com/paper/1811.01753/) (2018)
- [On the effectiveness of persistent homology](https://mlsanity.com/paper/2206.10551/) (2022)
- [Topological Data Analysis of Decision Boundaries with Application to Model Selection](https://mlsanity.com/paper/1805.09949/) (2018)
- [Differentiable Euler Characteristic Transforms for Shape Classification](https://mlsanity.com/paper/2310.07630/) (2023)
- [Euler Characteristic Tools For Topological Data Analysis](https://mlsanity.com/paper/2303.14040/) (2023)
- [Hypothesis Testing for Topological Data Analysis](https://mlsanity.com/paper/1310.7467/) (2013)
- [Euler Characteristic Curves and Profiles: a stable shape invariant for big data problems](https://mlsanity.com/paper/2212.01666/) (2022)
- [A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classification](https://mlsanity.com/paper/2605.02836/) (2026)
- [Statistical Inference for Persistence Diagrams via Landmark Embeddings: Minimax Theory and Finite Approximation](https://mlsanity.com/paper/2609.07691/) (2026)

## Built on by (most cited first, 1 of 1)

- [The Null Is the Hard Part: Exact Tests for Memorization in Generative Models](https://mlsanity.com/paper/2610.00251/) (2026)

## Abstract (arXiv)

Class disentanglement--the separation of a representation's class-conditional point clouds along depth and over training--is measured by descriptive curves: the sentence such a study wants to write, layer l+1 is more disentangled than layer l, is an eyeball judgement with no null. We supply the inferential layer for a topological measurement of class overlap, the Intersection Euler Characteristic Profile: the Euler characteristic of the overlap of the clouds' ball unions as a function of scale, from one Alpha-complex sweep with no boundary-matrix reduction. Every number carries a test--exact permutation tests in both directions, a guarded separation certificate the invariant requires, and a paired sign-flip test for comparative claims. Building that test taught a lesson outliving this invariant: its statistic must be scale-free. On the raw profile mass, which has units of feature length, 12,375 paired tests return 5,633 significant steps of which every one at the first epoch points the wrong way, certifying feature-norm dynamics as disentanglement; the dimensionless statistic returns 2,707, with 2,026 decreases. Across 111 networks and 52,650 measurements, disentanglement is depth-graded and early, and interaction quotients rank class pairs by confusability (rho=0.83), on par with cheap separability statistics. In a 96-model factorial, augmentation is the one training choice that separates classes relative to chance; weight decay compresses the overlap without separating. Only a k-fold statistic can pose the structural question: the joint entanglement of a class triple sits below its strongest pair in 97% of triple-layer cells and 99.5% of deep cells, at median ratios far below a measured null floor, in vision encoders and frozen language models--a regularity, not a law. The unnormalized mass predicts test accuracy (R^2=0.94), the quotient does not, and neither beats a linear probe.