# Function-Counting Theory for Low-Dimensional Data Structures

arXiv: https://arxiv.org/abs/2607.01010 · HTML: https://mlsanity.com/paper/2607.01010/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 29 references that are in the map.
- First posted on arXiv 1 Jul 2026.

## Builds on (most cited first, 1 of 1)

- [Separation Capacity of Scattering Networks](https://mlsanity.com/paper/2606.30822/) (2026)

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

- [Separation Capacity of Scattering Networks on Low-Dimensional Datasets](https://mlsanity.com/paper/2607.06048/) (2026)

## Abstract (arXiv)

The success of deep learning models in classification and regression is widely attributed to the low-dimensional structure that real-world data tend to exhibit, despite their high-dimensional representation. This work attempts to provide a mathematical framework for binary classification on low-dimensional data, building on Cover's (1965) function-counting theory. With our framework, we aim to address the question of how the low-dimensional structure of the data affects the classification capabilities of learning models. Cover's theory relies on a general position assumption that blinds it to the underlying data structure. We refine this assumption to account for the low-dimensionality of the data and derive dichotomy counts that reflect the data structure. We further extend Cover's separation capacity and problem of generalization to the low-dimensional setting, enabling the impact of the underlying data structure on both to be analyzed.