AI 中文总结
该研究针对双变量三分量高斯混合，构造出含至少7个模式的异方差混合密度，为高斯混合模式数猜想提供了首个反例，证明原上界不成立。
AI 中文摘要
高斯混合密度的模式数可多于其分量数。已有猜想称，d元k分量高斯混合密度的最大模式数为组合数C(d+k-1,d)，当(d,k)=(2,3)时该值为6。我们构造了一族等权重的异方差三分量双变量高斯混合密度，其至少包含7个不同的非退化模式，表明该猜想的上界对(d,k)=(2,3)不成立。据我们所知，这是所有(d,k)对中该猜想的首个反例。
英文摘要
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 $(d,k)$.
Comments10 pages, 3 figures
