论文标题
高斯人与统一的手段的混合物可以有几种模式？
How Many Modes Can a Mixture of Gaussians with Uniformly Bounded Means Have?
论文作者
论文摘要
我们通过明确的结构表明，单变量的高斯密度$ 1 $，并在$ [ - a，a，a] $中具有$ω（a^2）$模式的混合物。这反驳了最近对Dytso，Yagli，dofor和Shamai \ cite {dyps20}的猜想，他们表明这种混合物最多可以具有$ o（a^{2}）$模式，并推测上限可以将上限提高到$ o（a）$。即使对混合分布施加了额外的差异约束，我们的结果也成立。将结果扩展到更高的维度，我们在$ \ mathbb {r}^{d} $中展示了高斯人的混合物，并具有身份协方差，并在$ [ - a，a，a，a]^{d} $中，具有$ω（a^{2d}）$模式。
We show, by an explicit construction, that a mixture of univariate Gaussian densities with variance $1$ and means in $[-A,A]$ can have $Ω(A^2)$ modes. This disproves a recent conjecture of Dytso, Yagli, Poor and Shamai \cite{DYPS20} who showed that such a mixture can have at most $O(A^{2})$ modes and surmised that the upper bound could be improved to $O(A)$. Our result holds even if an additional variance constraint is imposed on the mixing distribution. Extending the result to higher dimensions, we exhibit a mixture of Gaussians in $\mathbb{R}^{d}$, with identity covariances and means inside $[-A,A]^{d}$, that has $Ω(A^{2d})$ modes.
