# Linear Classification of Neural Manifolds with Correlated Variability

arXiv: https://arxiv.org/abs/2211.14961 · HTML: https://mlsanity.com/paper/2211.14961/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 57 references that are in the map.
- First posted on arXiv 27 Nov 2022.

## Builds on (most cited first, 23 of 23)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [VICReg: Variance-Invariance-Covariance Regularization for Self-Supervised Learning](https://mlsanity.com/paper/2105.04906/) (2021)
- [Insights on representational similarity in neural networks with canonical correlation](https://mlsanity.com/paper/1806.05759/) (2018)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [A mathematical theory of semantic development in deep neural networks](https://mlsanity.com/paper/1810.10531/) (2018)
- [Classification and Geometry of General Perceptual Manifolds](https://mlsanity.com/paper/1710.06487/) (2017)
- [Neural population geometry: An approach for understanding biological and artificial neural networks](https://mlsanity.com/paper/2104.07059/) (2021)
- [Linear Readout of Object Manifolds](https://mlsanity.com/paper/1512.01834/) (2015)
- [Efficiency of local learning rules in threshold-linear associative networks](https://mlsanity.com/paper/2007.12584/) (2020)
- [Quality of internal representation shapes learning performance in feedback neural networks](https://mlsanity.com/paper/2011.06066/) (2020)
- [Robust information propagation through noisy neural circuits](https://mlsanity.com/paper/1608.05706/) (2016)
- [Learning Data Manifolds with a Cutting Plane Method](https://mlsanity.com/paper/1705.09944/) (2017)
- [Capacity of the covariance perceptron](https://mlsanity.com/paper/1912.00824/) (2019)
- [Geometric framework to predict structure from function in neural networks](https://mlsanity.com/paper/2010.09660/) (2020)
- [Capacity of Group-invariant Linear Readouts from Equivariant Representations: How Many Objects can be Linearly Classified Under All Possible Views?](https://mlsanity.com/paper/2110.07472/) (2021)
- [Soft-margin classification of object manifolds](https://mlsanity.com/paper/2203.07040/) (2022)

## Built on by (most cited first, 9 of 9)

- [The Geometry of Prompting: Unveiling Distinct Mechanisms of Task Adaptation in Language Models](https://mlsanity.com/paper/2502.08009/) (2025)
- [Neural population geometry and optimal coding of tasks with shared latent structure](https://mlsanity.com/paper/2402.16770/) (2024)
- [Feature Learning beyond the Lazy-Rich Dichotomy: Insights from Representational Geometry](https://mlsanity.com/paper/2503.18114/) (2025)
- [Inversion dynamics of class manifolds in deep learning reveals tradeoffs underlying generalisation](https://mlsanity.com/paper/2303.05161/) (2023)
- [Probing Biological and Artificial Neural Networks with Task-dependent Neural Manifolds](https://mlsanity.com/paper/2312.14285/) (2023)
- [A statistical physics framework for optimal learning](https://mlsanity.com/paper/2507.07907/) (2025)
- [Diagnosing Generalization Failures from Representational Geometry Markers](https://mlsanity.com/paper/2603.01879/) (2026)
- [Can Stationary Distributions of Scale-Invariant Neural Networks Be Described by the Thermodynamics of an Ideal Gas?](https://mlsanity.com/paper/2511.07308/) (2025)
- [Sparse Prototype Code Underlies Classification and Prediction Across Modalities](https://mlsanity.com/paper/2608.15632/) (2026)

## Abstract (arXiv)

Understanding how the statistical and geometric properties of neural activity relate to performance is a key problem in theoretical neuroscience and deep learning. Here, we calculate how correlations between object representations affect the capacity, a measure of linear separability. We show that for spherical object manifolds, introducing correlations between centroids effectively pushes the spheres closer together, while introducing correlations between the axes effectively shrinks their radii, revealing a duality between correlations and geometry with respect to the problem of classification. We then apply our results to accurately estimate the capacity of deep network data.