Obtain sharp mode and critical-point bounds for isotropic Gaussian mixtures

Determine, for every pair \((n,d)\), the sharp upper bounds on the numbers of critical points, modes, and global modes of an \(n\)-component isotropic Gaussian mixture in dimension \(d\).

Background

The paper proves qualitative finiteness of the critical set for every finite isotropic Gaussian mixture, thereby making previously conditional quantitative mode-counting bounds applicable. However, the functional-transcendence proof is non-effective and does not itself give an upper bound on the number of critical points or modes.

Existing general bounds are described as far from sharp, and the paper therefore explicitly asks for the exact extremal quantities as functions of the number of components nn and ambient dimension dd. The author also states that no existing conjecture predicts these sharp bounds.

References

For each (n,d), what are the sharp upper bounds on the numbers of critical points, modes, and global modes of an n-component isotropic Gaussian mixture in dimension d? To the best of the author's knowledge, no conjecture currently addresses these sharp bounds.

— On Finite Gaussian Mixtures: Finiteness of the Number of Modes and an Application to NPMLE  (2608.16675 - Wang, 17 Aug 2026) in Question in the paragraph “Quantitative mode counting,” Section 4, “Discussion”

While the finiteness of $m(d,k)$ is now established, determining its value remains open.

\begin{problem}[The maximal Gaussian mode problem] Determine $m(d,k)$, or obtain upper and lower bounds that are sharp in their dependence on $d$ and $k$. \end{problem}

— A ridgeline correspondence criterion: the number of modes of a Gaussian mixture is finite  (2608.28558 - Améndola et al., 28 Aug 2026) in Problem (The maximal Gaussian mode problem), Section Discussion and conclusion, subsection “The maximum number of modes as an open problem”