多项式时间内固定分量高斯位置混合模型的精确恰当估计
Sharp proper estimation of fixed-component Gaussian location mixtures in polynomial time
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中文总结 AI 辅助
本研究解决固定\\(k\geq3\\)时多项式时间内精确估计高斯位置混合模型的问题，通过矩纤维范围查找器等工具实现最优Hellinger速率。
中文摘要 AI 辅助
我们考虑\\(\mathbb{R}^d\\)中最多包含\\(k\\)个单位协方差高斯分布的混合模型，其均值属于固定半径球，无分离或最小权重条件。Doss、Wu、Yang和Zhou（2023）证明了极小极大Hellinger风险的阶为\\(\sqrt{d/n}\wedge 1\\)，并构造了一个恰当多项式时间估计器，其通用界为较慢的\\((d/n)^{1/4}\\)；对于固定\\(k\geq 3\\)，在多项式时间内获得精确速率的问题仍未解决。我们解决了该问题。关键工具是矩纤维范围查找器。二阶矩子空间控制投影遗漏的能量。随后我们估计有限个单自由索引的Hermite收缩量。这些向量值收缩量以精确的\\(\sqrt{d/n}\\)尺度恢复了恰好包含一个遗漏方向的每个张量分量。其余每个项至少包含两个遗漏因子，因此由残差二阶矩能量控制。所得子空间的维度仅依赖于\\(k\\)。在该常数维空间中进行穷举矩拟合可生成一个恰当混合模型，结合高斯混合模型的无维度矩表征，在多项式算术时间内实现了每个固定\\(k\\)的最优Hellinger速率。
英文摘要
We consider a mixture of at most $k$ unit-covariance Gaussians in $\mathbb{R}^d$ whose means belong to a fixed-radius ball, with no separation or minimum-weight condition. Doss, Wu, Yang and Zhou (2023) proved that the minimax Hellinger risk is of order $\sqrt{d/n}\wedge 1$ and constructed a proper polynomial-time estimator with the slower general bound $(d/n)^{1/4}$; obtaining the sharp rate in polynomial time for fixed $k\geq 3$ was left open. We resolve this question. The key device is a moment-fiber range finder. A second-moment subspace controls the energy missed by projection. We then estimate finitely many one-free-index Hermite contractions. These vector-valued contractions recover every tensor component containing exactly one missed direction at the sharp $\sqrt{d/n}$ scale. Every remaining term contains at least two missed factors and is therefore controlled by the residual second-moment energy. The resulting subspace has dimension depending only on $k$. 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 $k$.
