How many modes can a constrained Gaussian mixture have?

05/04/2020

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by   Navin Kashyap, et al.

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We show, by an explicit construction, that a mixture of univariate Gaussians 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 [IEEE Trans. Inform. Theory, Apr. 2020], 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 R^d, with identity covariances and means inside [-A,A]^d, that has Ω(A^2d) modes.