论文标题
通过量子扩展为Tyler的M估计器进行严格保证
Rigorous Guarantees for Tyler's M-estimator via quantum expansion
论文作者
论文摘要
估计椭圆形分布的形状是统计中的一个基本问题。泰勒（Tyler）的M-估计器的形状矩阵的一个估计量已被证明具有许多吸引人的渐近特性。它在数值实验中表现良好，可以通过简单的迭代程序快速计算。尽管估计量已经在统计界进行了多年的研究，但估计速率的速度既不紧密，也没有任何迭代程序在多个哪个步骤中收敛的证据。 在这里，我们观察到Tyler的M-估计器与操作员缩放之间存在令人惊讶的联系，近年来，它已经进行了深入研究，部分原因是它与Brascamp-Lieb分析中的不平等现象有关。我们将这种连接与量子扩张器上的新结果一起使用，以表明Tyler的M估计器具有最佳的速率，直至尺寸中的对数因子对数，并且在生成模型中，迭代过程即使没有正则化也具有线性收敛速率。
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.
