How Many Modes Can a Mixture of Gaussians with Uniformly Bounded Means\n Have?

2020/05/04 by Navin Kashyap, Kashyap, Navin, Manjunath Krishnapur +1
Chemistry · Decision Sciences · Mathematics · #Advanced Statistical Methods and Models #Advanced Statistical Process Monitoring #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Spectroscopy and Chemometric Analyses #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2005.01580

openalex publication_date 2020/05/04 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We show, by an explicit construction, that a mixture of univariate Gaussian\ndensities with variance 1 and means in [-A,A] can have \Ω(A²) modes.\nThis disproves a recent conjecture of Dytso, Yagli, Poor and Shamai\n citeDYPS20 who showed that such a mixture can have at most O(A²) modes\nand surmised that the upper bound could be improved to O(A). Our result holds\neven if an additional variance constraint is imposed on the mixing\ndistribution. Extending the result to higher dimensions, we exhibit a mixture\nof Gaussians in \ℝ^d, with identity covariances and means inside\n[-A,A]^d, that has \Ω(A^2d) modes.\n

Related

• On Mixtures of Three Homoscedastic Gaussian Densities: An Unconditional Sharper Bound on the Number of Modes
• Learning Gaussian Mixtures with Arbitrary Separation
• Tight bounds for learning a mixture of two gaussians
• Learning Mixtures of Gaussians in High Dimensions
• Mixture Models, Robustness, and Sum of Squares Proofs