# Soft-margin classification of object manifolds

arXiv: https://arxiv.org/abs/2203.07040 · HTML: https://mlsanity.com/paper/2203.07040/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 24 references that are in the map.
- First posted on arXiv 14 Mar 2022.

## Builds on (most cited first, 5 of 5)

- [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)
- [Statistical Mechanics of High-Dimensional Inference](https://mlsanity.com/paper/1601.04650/) (2016)
- [Linear Readout of Object Manifolds](https://mlsanity.com/paper/1512.01834/) (2015)
- [Learning Data Manifolds with a Cutting Plane Method](https://mlsanity.com/paper/1705.09944/) (2017)

## Built on by (most cited first, 2 of 2)

- [Linear Classification of Neural Manifolds with Correlated Variability](https://mlsanity.com/paper/2211.14961/) (2022)
- [Dynamical stability for dense patterns in attractor neural networks](https://mlsanity.com/paper/2507.10383/) (2025)

## Abstract (arXiv)

A neural population responding to multiple appearances of a single object defines a manifold in the neural response space. The ability to classify such manifolds is of interest, as object recognition and other computational tasks require a response that is insensitive to variability within a manifold. Linear classification of object manifolds was previously studied for max-margin classifiers. Soft-margin classifiers are a larger class of algorithms and provide an additional regularization parameter used in applications to optimize performance outside the training set by balancing between making fewer training errors and learning more robust classifiers. Here we develop a mean-field theory describing the behavior of soft-margin classifiers applied to object manifolds. Analyzing manifolds with increasing complexity, from points through spheres to general manifolds, a mean-field theory describes the expected value of the linear classifier's norm, as well as the distribution of fields and slack variables. By analyzing the robustness of the learned classification to noise, we can predict the probability of classification errors and their dependence on regularization, demonstrating a finite optimal choice. The theory describes a previously unknown phase transition, corresponding to the disappearance of a non-trivial solution, thus providing a soft version of the well-known classification capacity of max-margin classifiers.