From Gauss to Kolmogorov: Localized Measures of Complexity for Ellipses
From Gauss to Kolmogorov: Localized Measures of Complexity for Ellipses
复制标题
从高斯到柯尔莫哥洛夫：椭圆复杂性的局部度量
DOI：
--
复制
发表时间：
2018
影响因子：
1.1
通讯作者：
M. Wainwright
中科院分区：
文献类型：
--
作者：
Yuting Wei;Billy Fang;M. Wainwright
高斯宽度是概率、统计和几何中的一个基本量，众所周知，它是估计和假设检验的内在困难的基础。在这项工作中，我们展示了当定位到椭圆的任何给定点时，高斯宽度如何可以通过类似定位的集合的柯尔莫哥洛夫宽度来控制。这种联系导致最小二乘回归的估计误差明确表征为椭圆内真实回归向量的函数。误差衰减率随着位置的变化而变化很大：作为一个具体的例子，在光滑度 $alpha$ 的 Sobolev 椭圆中，我们表现出的速率从对应于经典全局速率的 $(sigma^2)^{frac{2 alpha}{2 alpha + 1}}$ 到更快的速率 $(sigma^2)^{frac{4 alpha}{4 alpha + 1}}$ 不等。我们还展示了局部柯尔莫哥洛夫宽度如何与局部度量熵相关。
The Gaussian width is a fundamental quantity in probability, statistics and geometry, known to underlie the intrinsic difficulty of estimation and hypothesis testing. In this work, we show how the Gaussian width, when localized to any given point of an ellipse, can be controlled by the Kolmogorov width of a set similarly localized. This connection leads to an explicit characterization of the estimation error of least-squares regression as a function of the true regression vector within the ellipse. The rate of error decay varies substantially as a function of location: as a concrete example, in Sobolev ellipses of smoothness $alpha$, we exhibit rates that vary from $(sigma^2)^{frac{2 alpha}{2 alpha + 1}}$, corresponding to the classical global rate, to the faster rate $(sigma^2)^{frac{4 alpha}{4 alpha + 1}}$. We also show how the local Kolmogorov width can be related to local metric entropy.