arXiv subjects

Hirotaka Matsumoto

Publications and source records attributed to Hirotaka Matsumoto.

2 recordsLinked to original sources

arXiv2026

At least seven modes in a heteroscedastic three-component bivariate Gaussian mixture

A Gaussian mixture density can have more modes than components. It has been conjectured that the maximum number of modes of a $d$-variate $k$-component Gaussian mixture density is $\binom{d+k-1}{d}$, which equals six for $(d,k)=(2,3)$. We construct an explicit family of equally weighted heteroscedastic three-component bivariate Gaussian mixture densities with at least seven distinct nondegenerate modes, showing that this conjectured upper bound fails for $(d,k)=(2,3)$. To the best of our knowledge, this provides the first counterexample to the conjecture across all pairs $(d,k)$.

math.ST↗

arXiv2025

Singularities in bivariate normal mixtures

We investigate mappings $F = (f_1, f_2) \colon \mathbb{R}^2 \to \mathbb{R}^2 $ where $ f_1, f_2 $ are bivariate normal densities from the perspective of singularity theory of mappings, motivated by the need to understand properties of two-component bivariate normal mixtures. We show a classification of mappings $ F = (f_1, f_2) $ via $\mathcal{A}$-equivalence and characterize them using statistical notions. Our analysis reveals three distinct types, each with specific geometric properties. Furthermore, we determine the upper bounds for the number of modes in the mixture for each type.

math.ST↗