AI 中文总结
该研究证明有限分量的各向同性高斯混合模型的临界点集有限，将其应用于高斯位置混合模型，得出基于有限数据集的NPMLE具有有限支撑且所有NPMLE共享相同可允许原子位置的结论。
AI 中文摘要
我们证明了每个具有有限个分量的各向同性高斯混合模型都具有有限个模态。在一维情况下，切比雪夫系统的经典理论给出了n分量混合模型最多具有n个模态的精确界。然而在高维情况下，这类混合模型是否都具有有限个模态一直是未解决的问题。我们的主要结果是更强的结论：整个临界点集具有有限基数，该结论通过结合实解析曲线选择定理与Ax的函数超越性定理得到证明。作为应用，我们表明对于高斯位置混合模型，基于有限数据集的每个非参数极大似然估计（NPMLE）都具有有限支撑；更具体地说，所有NPMLE都共享相同的有限个可允许原子位置集合。
英文摘要
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 finitely many modes. Our main result is the stronger statement that the entire critical set has finite cardinality, which is proven by combining real analytic curve selection theorem and Ax's functional-transcendence theorem. As an application, we show that, for Gaussian location mixtures, every nonparametric maximum likelihood estimator (NPMLE) based on a finite dataset is finitely supported. More specifically, all NPMLEs share the same finite set of allowable atom locations.
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