Three Homoscedastic Gaussian Components Have at Most Four Modes
Abstract
Gaussian mixtures can have more modes than components. For three Gaussian components with a common covariance, four modes are attainable, whereas the best general upper bound was eight. We prove that four is the exact maximum in every dimension d ≥ 2, for arbitrary means and positive weights. We further show that the sharp bound drops to three when the whitened means are collinear or form a right or obtuse triangle. The result covers degenerate modes without genericity or separate finiteness assumptions. Our proof maps modes to local minima of a separable free-energy functional on the probability simplex, reduces the problem to counting scalar inverse-function branches, and controls competing branches through curvature and stationary-height comparisons.
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BibTeX
@misc{okahara2026three,
title = {{Three Homoscedastic Gaussian Components Have at Most Four Modes}},
author = {Rintaro Okahara and Kei Kobayashi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15369},
eprint = {2610.15369},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15369},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Three Homoscedastic Gaussian Components Have at Most Four Modes
AU - Rintaro Okahara
AU - Kei Kobayashi
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15369
UR - https://arxiv.org/abs/2610.15369
AB - Gaussian mixtures can have more modes than components. For three Gaussian components with a common covariance, four modes are attainable, whereas the best general upper bound was eight. We prove that four is the exact maximum in every dimension d ≥ 2, for arbitrary means and positive weights. We further show that the sharp bound drops to three when the whitened means are collinear or form a right or obtuse triangle. The result covers degenerate modes without genericity or separate finiteness assumptions. Our proof maps modes to local minima of a separable free-energy functional on the probability simplex, reduces the problem to counting scalar inverse-function branches, and controls competing branches through curvature and stationary-height comparisons.
ER -