A ridgeline correspondence criterion: the number of modes of a Gaussian mixture is finite
Abstract
We prove that every finite multivariate Gaussian mixture density has only finitely many modes. Our approach combines an algebraic formulation of the ridgeline theory of Ray and Lindsay (2005) with a transcendence-degree argument based on Ax's functional-transcendence theorem to bound the cardinality of the set of critical points. Our techniques extend recent work by Wang (2026), who used Ax's theorem together with real-analytic curve selection to prove finiteness of the critical set of homoscedastic Gaussian mixtures. We introduce the ridgeline correspondence and use it to obtain a finiteness result that applies to arbitrary heteroscedastic Gaussian mixtures. Our framework also establishes finiteness of the number of modes for additional classes of polynomial-exponential mixtures and generalized Gaussian mixtures.
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BibTeX
@misc{amendola2026ridgeline,
title = {{A ridgeline correspondence criterion: the number of modes of a Gaussian mixture is finite}},
author = {Carlos Améndola and Jose Israel Rodriguez},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28558},
eprint = {2608.28558},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28558},
}
RIS
TY - UNPB
TI - A ridgeline correspondence criterion: the number of modes of a Gaussian mixture is finite
AU - Carlos Améndola
AU - Jose Israel Rodriguez
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28558
UR - https://arxiv.org/abs/2608.28558
AB - We prove that every finite multivariate Gaussian mixture density has only finitely many modes. Our approach combines an algebraic formulation of the ridgeline theory of Ray and Lindsay (2005) with a transcendence-degree argument based on Ax's functional-transcendence theorem to bound the cardinality of the set of critical points. Our techniques extend recent work by Wang (2026), who used Ax's theorem together with real-analytic curve selection to prove finiteness of the critical set of homoscedastic Gaussian mixtures. We introduce the ridgeline correspondence and use it to obtain a finiteness result that applies to arbitrary heteroscedastic Gaussian mixtures. Our framework also establishes finiteness of the number of modes for additional classes of polynomial-exponential mixtures and generalized Gaussian mixtures.
ER -