2026/08/03 by Yutaro Kabata, Hirotaka Matsumoto, Akifumi Okuno
Mathematics · #math.ST #stat.TH
10 pages, 3 figures
arxiv created 2026/08/03 · arxiv updated 2026/08/04
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 \binomd+k-1d, 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).