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.
翻译:估测椭圆分布的形状是统计中的一个根本问题。 形状矩阵的一个估计者, Tyler 的 M- 估计器, 已证明它有许多有吸引力的无药可依的特性。 它在数字实验中表现良好, 并且可以通过简单的迭接程序在实践上快速计算。 尽管统计界已经研究过这个估计器多年, 但没有严格的非无药可依地限制测量器的速率, 也没有证明迭接程序在多步之间会融合。 我们在这里观察到泰勒的M- 估计器和操作器的缩放之间有惊人的连接, 这一点近年来已经深入研究过, 部分原因是它与Brascamp- Lieb的不平等性在分析中存在联系。 我们使用这个连接, 加上量子扩张器的新结果, 以显示泰勒的M- 估计器的速率最符合维度的对数系数, 而在基因组化模型中, 迭接合程序有线性趋同率, 即使没有正规化。