# Directional Linear Separability of Neural Representations: Geometry and Transformations

arXiv: https://arxiv.org/abs/2606.08721 · HTML: https://mlsanity.com/paper/2606.08721/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 36 references that are in the map.
- First posted on arXiv 7 Jun 2026.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Decoupled Weight Decay Regularization](https://mlsanity.com/paper/1711.05101/) (2017)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Emergence of Separable Manifolds in Deep Language Representations](https://mlsanity.com/paper/2006.01095/) (2020)
- [Separation and Concentration in Deep Networks](https://mlsanity.com/paper/2012.10424/) (2020)
- [Randomly Initialized One-Layer Neural Networks Make Data Linearly Separable](https://mlsanity.com/paper/2205.11716/) (2022)
- [Linearly Separable Features in Shallow Nonlinear Networks: Width Scales Polynomially with Intrinsic Data Dimension](https://mlsanity.com/paper/2501.02364/) (2025)

## Abstract (arXiv)

Neural networks build representations through affine maps and nonlinear activations. Injective affine maps preserve linear separability, raising the problem of how they prepare data for nonlinear improvement and how much gain can be guaranteed before complete separation. We introduce the directional linear separability measure (D-LSM), which quantifies unavoidable competing-sample intrusion over affine halfspaces retaining every target sample, characterize its supporting geometry, and prove invariance under injective affine embeddings. For gated activations including ReLU, GELU, and SiLU, pre-activation projection bounds yield sufficient conditions for preserving all previous exclusions and recovering additional samples, with a gain bound determined by the certified recovery count. Under an aggregate-tube condition, an explicit affine construction realizes recovery with sufficient width, scaling conditions, and simultaneous multiclass guarantees through a shared layer. Exact controlled experiments compare certified and realized gains, assess certificate coverage, and exhibit bound attainment before complete separation and in affine-tube constructions. In learned Vision Transformer (ViT) representations, a feasible lower-bound estimator yields earlier post-GELU saturation certificates of exact separability, while boundary transport numerically supports affine invariance.