Tyler M-估计量的量子展开严格保证
Rigorous Guarantees for Tyler's M-estimator via quantum expansion
摘要：
估计椭圆分布的形状是统计学中的一个基本问题。形状矩阵的一个估计，泰勒的M-估计，已被证明有许多吸引人的渐近性质。它在数值实验中表现良好，并且可以通过简单的迭代过程在实践中快速计算。尽管多年来的估计已被研究的统计社区，既没有一个严格的非渐近界的估计率，也没有证明迭代过程收敛于多项式的许多步骤。在这里，我们观察到一个令人惊讶的连接泰勒的M-估计和运营商标度，这已被深入研究，在最近几年的部分原因是它的连接到Brascamp-Lieb不等式分析。我们使用这种连接，再加上新的结果，量子扩展，表明泰勒的M-估计的最佳速率对数的因素的维度，并在生成模型的迭代过程中，即使没有正则化的线性收敛速度。
英文摘要：
Estimating the shape of an elliptical distribution is a fundamental problem in statistics. One estimator for the shape matrix, Tyler's M-estimator, has been shown to have many appealing asymptotic properties. It performs well in numerical experiments and can be quickly computed in practice by a simple iterative procedure. Despite the many years the estimator has been studied in the statistics community, there was neither a tight non-asymptotic bound on the rate of the estimator nor a proof that the iterative procedure converges in polynomially many steps. Here we observe a surprising connection between Tyler's M-estimator and operator scaling, which has been intensively studied in recent years in part because of its connections to the Brascamp-Lieb inequality in analysis. We use this connection, together with novel results on quantum expanders, to show that Tyler's M-estimator has the optimal rate up to factors logarithmic in the dimension, and that in the generative model the iterative procedure has a linear convergence rate even without regularization.
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。
