# Maximum Number of Modes of Gaussian Mixtures

## TL;DR

This paper investigates the maximum number of modes in Gaussian mixture models, providing new lower bounds and the first finite upper bound, advancing understanding of their complex density landscapes.

## Contribution

It introduces improved lower bounds and the first finite upper bound on the number of modes in Gaussian mixtures, addressing a longstanding open problem.

## Key findings

- Established new lower bounds on modes
- Derived the first finite upper bound on modes
- Enhanced understanding of Gaussian mixture density complexity

## Abstract

Gaussian mixture models are widely used in Statistics. A fundamental aspect of these distributions is the study of the local maxima of the density, or modes. In particular, it is not known how many modes a mixture of $k$ Gaussians in $d$ dimensions can have. We give a brief account of this problem's history. Then, we give improved lower bounds and the first upper bound on the maximum number of modes, provided it is finite.