# Stochastic Separability of Embedding Manifolds

arXiv: https://arxiv.org/abs/2608.22874 · HTML: https://mlsanity.com/paper/2608.22874/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 25 references that are in the map.
- First posted on arXiv 24 Aug 2026.

## Builds on (most cited first, 9 of 9)

- [Classification and Geometry of General Perceptual Manifolds](https://mlsanity.com/paper/1710.06487/) (2017)
- [Blessing of dimensionality: mathematical foundations of the statistical physics of data](https://mlsanity.com/paper/1801.03421/) (2018)
- [Neural population geometry: An approach for understanding biological and artificial neural networks](https://mlsanity.com/paper/2104.07059/) (2021)
- [Estimating the effective dimension of large biological datasets using Fisher separability analysis](https://mlsanity.com/paper/1901.06328/) (2019)
- [Correction of AI systems by linear discriminants: Probabilistic foundations](https://mlsanity.com/paper/1811.05321/) (2018)
- [High-dimensional separability for one- and few-shot learning](https://mlsanity.com/paper/2106.15416/) (2021)
- [Stochastic Separation Theorems](https://mlsanity.com/paper/1703.01203/) (2017)
- [General stochastic separation theorems with optimal bounds](https://mlsanity.com/paper/2010.05241/) (2020)
- [High-dimensional brain. A tool for encoding and rapid learning of memories by single neurons](https://mlsanity.com/paper/1710.11227/) (2017)

## Abstract (arXiv)

Neurobiological studies and representation learning have observed that representations of objects belonging to the same category in high-dimensional neural spaces exhibit low-dimensional object manifold characteristics, and different object manifolds are linearly separable in these neural spaces. However, these experimentally observed phenomena lack rigorous theoretical validation to date. This paper proposes a new stochastic separability theorem for embedding manifolds of two different object categories. First, we establish a projection measure concentration theorem for embedding manifolds under general conditions. We develop a new two-layer measure concentration analysis technique, which unifies two estimation bounds via the law of total expectation to derive measure concentration inequalities. Based on the measure concentration theorem, we further prove a stochastic separability theorem for embedding manifolds of two different object categories. If two datasets have distinct means and bounded total variances, their samples become linearly separable with high probability, provided that the projection direction satisfies a non-singularity condition. The main contributions of this paper are twofold: 1. We prove the projection concentration properties of embedding manifolds in high-dimensional spaces by using two-lawyer tail-bound inequalities. 2. We identify a non-singularity condition for the stochastic separability between embedding manifolds, and rigorously prove the stochastic projection separability theorem. The theorem not only uncovers geometric and statistical properties of the object embedding manifolds, but also provides a novel mechanism for representation learning in deep networks.