M-方差的渐近性质与描述符的唯一性
M-Variance Asymptotics and Uniqueness of Descriptors
AI总结：
本文为M-估计中的样本损失最小值（M-方差）构建了渐近理论，利用其满足中心极限定理的特性提出假设检验，以判断总体损失函数是否存在多个全局最小值，并验证了其在非欧空间均值、非线性回归和高斯混合聚类中的应用。
摘要：
M-估计问题的渐近理论通常关注样本描述符的渐近收敛性，样本描述符被定义为样本损失函数的最小化点。在此，我们探讨一个相关问题，并为样本损失的最小值（即M-方差）构建渐近理论。由于损失函数值始终为实数，M-方差的渐近理论相对简单。M-方差通常满足标准的中心极限定理，即使在描述符渐近性质更复杂的情况下（例如在smeariness的情况下），或者在无法给出渐近分布的情况下（例如描述符空间为一般度量空间时），也是如此。我们利用M-方差的渐近结果构建了一个假设检验，以系统地确定对于给定样本，底层总体损失函数是否可能具有多个全局最小值。我们讨论了该方法在数据中的三个应用，每个应用都展示了描述符非唯一性可能发生的典型场景。这些模型场景包括非欧几里得空间上的均值、非线性回归和高斯混合聚类。
英文摘要：
Asymptotic theory for M-estimation problems usually focuses on the asymptotic convergence of the sample descriptor, defined as the minimizer of the sample loss function. Here, we explore a related question and formulate asymptotic theory for the minimum value of sample loss, the M-variance. Since the loss function value is always a real number, the asymptotic theory for the M-variance is comparatively simple. M-variance often satisfies a standard central limit theorem, even in situations where the asymptotics of the descriptor is more complicated as for example in case of smeariness, or if no asymptotic distribution can be given as can be the case if the descriptor space is a general metric space. We use the asymptotic results for the M-variance to formulate a hypothesis test to systematically determine for a given sample whether the underlying population loss function may have multiple global minima. We discuss three applications of our test to data, each of which presents a typical scenario in which non-uniqueness of descriptors may occur. These model scenarios are the mean on a non-euclidean space, non-linear regression and Gaussian mixture clustering.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
