Function-Counting Theory for Low-Dimensional Data Structures

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.