# Mean Dimension of Ridge Functions

## TL;DR

This paper investigates how the mean dimension of ridge functions behaves in high dimensions, showing that Lipschitz continuity keeps it bounded, while discontinuities can cause it to grow with dimension, and preintegration can reduce this growth.

## Contribution

It provides a detailed analysis of the mean dimension of ridge functions, highlighting the effects of Lipschitz continuity, sparsity, and preintegration on high-dimensional behavior.

## Key findings

- Lipschitz ridge functions have bounded mean dimension as dimension grows.
- Discontinuous ridge functions can have mean dimension proportional to .
- Preintegration can significantly reduce mean dimension from (}) to (1).

## Abstract

We consider the mean dimension of some ridge functions of spherical Gaussian random vectors of dimension $d$. If the ridge function is Lipschitz continuous, then the mean dimension remains bounded as $d\to\infty$. If instead, the ridge function is discontinuous, then the mean dimension depends on a measure of the ridge function's sparsity, and absent sparsity the mean dimension can grow proportionally to $\sqrt{d}$. Preintegrating a ridge function yields a new, potentially much smoother ridge function. We include an example where, if one of the ridge coefficients is bounded away from zero as $d\to\infty$, then preintegration can reduce the mean dimension from $O(\sqrt{d})$ to $O(1)$.