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 critical points. Our techniques extend recent work by Wang (2026), who used Ax's theorem together with real-analytic curve selection to prove finiteness of the critical set of homoscedastic Gaussian mixtures. We introduce the ridgeline correspondence and use it to obtain a finiteness result that applies to arbitrary heteroscedastic Gaussian mixtures. Our framework also establishes finiteness of the number of modes for additional classes of polynomial-exponential mixtures and generalized Gaussian mixtures.
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...
For finite-dimensional linear inverse problems where the variables are Gaussian, it is well-known that the minimum-mean-square error estimator takes the form of a regularized least-squares data fit. In this chapter, we show that this equivalence extends to a much broader infinite-dimensional setting where generalized s...
We determine the frame set of the first Hermite function, thereby proving a conjecture of Lyubarskii and Nes. A characterization of semi-regular Gabor frames due to Gr\"ochenig, Romero, and St\"ockler reduces the problem to a uniqueness question for entire functions in a Gaussian shift-invariant space. We address this...
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 a dimension-free Poincar\'e inequality for isotropic unconditional log-concave probability measures, establishing the Kannan-Lov\'asz-Simonovits (KLS) conjecture in the unconditional setting. The proof combines a new application of Dunkl operator theory combined with an appropriate transformation of the log-co...
In [Jalowy, Kabluchko, Marynych, Part I of the present work, Constr. Approx.], the authors discuss a user-friendly approach to determine the limiting empirical zero distribution of a sequence of real-rooted polynomials, as the degree goes to
$$\infty $$
∞
. In this note, we aim to apply it to a vast...
Jonas Jalowy, Zakhar Kabluchko, A. Marynych· Constructive approximation· 0 citations
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