AI 中文总结
该研究通过结合脊线理论代数表述与Ax函数超越定理，证明有限多元高斯混合等类混合模型的模态数有限，扩展了同方差高斯混合的相关有限性结果。
AI 中文摘要
我们证明，每个有限多元高斯混合密度仅具有有限个模态。我们的方法结合了Ray和Lindsay（2005）的脊线理论的代数表述，以及基于Ax函数超越定理的超越次数论证，以限制临界点集合的基数。我们的技术扩展了Wang（2026）的近期工作，Wang使用Ax定理结合实解析曲线选择证明了同方差高斯混合的临界点集合的有限性。我们引入脊线对应并将其用于获得适用于任意异方差高斯混合的有限性结果。我们的框架还为其他类别的多项式-指数混合和广义高斯混合建立了模态数的有限性。
英文摘要
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.
Comments22 pages, 2 figures
