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 $(d,k)$.
Exhaustive moment fitting in this constant-dimensional space produces a proper mixture and, together with the dimension-free moment characterization of Gaussian mixtures, achieves the optimal Hellinger rate in polynomial arithmetic time for every fixed $k$.
We prove that every isotropic Gaussian mixture with finitely many components has finitely many modes. In one dimension, classical theory of Chebyshev systems gives the sharp bound of at most $n$ modes for an $n$-component mixture. In several dimensions, however, it has remained open whether every such mixture has finit...
Let $q_n(r)$ denote the tail probability at $r$ of the self-normalized sum of $n$ independent centered uniform variables. At $r=3$, the first distribution-sensitive term in the two-sided Edgeworth expansion of Student's statistic vanishes. We evaluate the expansion at the common moving boundary $r_n=3+\lambda/n$ in dim...
We prove that every finite multivariate Gaussian mixture density has only finitely many modes. Our approach combines an algebraic formulation of the ridgeline theory of Ray and Lindsay (2005) with a transcendence-degree argument based on Ax's functional-transcendence theorem to bound the cardinality of the set of criti...
In this paper, the exact joint density of the top $k$ sample eigenvectors is derived for any $p\times p$ population covariance matrix $\varSigma$, sample size $n>p -1$, and $1 \le k \le p$. Using symmetric functions such as zonal polynomials and a dual summation identity by I. G. Macdonald, the result is expressed as a...
We study initialization bias in spherical $K$-means for weakly informative directional mixtures. We model the observations by a $K$-component von Mises-Fisher mixture with a small concentration parameter $\kappa$, corresponding to a high-dispersion regime in which the data provide limited information about the underlyi...
Amnon Balanov, Tamir Bendory· 0 citations
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