The Local Geometry of Testing in Ellipses: Tight Control via Localized Kolmogorov Widths
The Local Geometry of Testing in Ellipses: Tight Control via Localized Kolmogorov Widths
复制标题
椭圆测试的局部几何：通过局部柯尔莫哥洛夫宽度进行严格控制
DOI：
10.1109/tit.2020.2981313
复制
发表时间：
2017
影响因子：
2.5
通讯作者：
M. Wainwright
中科院分区：
文献类型：
--
作者：
Yuting Wei;M. Wainwright
椭圆上的均值测试是一个决策问题，出现在几个应用中，包括非参数拟合优度测试，认知无线电中的信号检测，以及再生核希尔伯特空间中的回归函数测试。我们研究了这种测试问题的局部几何与复合替代品。给定高斯随机向量的样本，目标是区分均值是否等于椭圆内的已知向量，或等于椭圆中的其他未知向量。虽然过去对这些问题的工作集中在一个全球意义上的困难，我们研究困难的方式，是本地化的椭圆内的每个向量。我们的主要结果是给尖锐的上界和下界的局部极小极大检验半径的一个明确的公式，涉及的Kolmogorov宽度的椭圆的椭圆形的Euclidean球。当应用于特定的例子，我们的一般定理产生有趣的利率是不知道的：作为一个特定的情况下，在索博列夫椭圆的光滑度<inline-formula><tex-math notation="LaTeX">$\alpha $</tex-math></inline-formula>的测试，我们展示了从<inline-formula><tex-math notation="LaTeX">$（\sigma ^{2}）^{\frac {4\alpha }{4 \alpha + 1}}$</tex-math></inline-formula>，对应于经典的全球利率，更快的速度<inline-formula><tex-math notation="LaTeX">$（\sigma ^{2}）^{\frac {8\alpha }{8 \alpha + 1}}$</tex-math></inline-formula>，可实现的矢量在椭圆内的有利位置。我们还表明，这个问题的最佳测试是通过线性投影测试，是基于一个明确的低维投影的观测向量。
Mean testing over ellipses is a decision problem that arises in several applications, including non-parametric goodness-of-fit testing, signal detection in cognitive radio, and regression function testing in reproducing kernel Hilbert spaces. We study the local geometry of such testing problems with compound alternatives. Given samples of a Gaussian random vector, the goal is to distinguish whether the mean is equal to a known vector within an ellipse, or equal to some other unknown vector in the ellipse. While past work on such problems has focused on the difficulty in a global sense, we study difficulty in a way that is localized to each vector within the ellipse. Our main result is to give sharp upper and lower bounds on the localized minimax testing radius in terms of an explicit formula involving the Kolmogorov width of the ellipse intersected with a Euclidean ball. When applied to particular examples, our general theorems yield interesting rates that were not known before: as a particular case, for testing in Sobolev ellipses of smoothness <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>, we demonstrate rates that vary from <inline-formula> <tex-math notation="LaTeX">$(\sigma ^{2})^{\frac {4 \alpha }{4 \alpha + 1}}$ </tex-math></inline-formula>, corresponding to the classical global rate, to the faster rate <inline-formula> <tex-math notation="LaTeX">$(\sigma ^{2})^{\frac {8 \alpha }{8 \alpha + 1}}$ </tex-math></inline-formula>, achievable for vectors at favorable locations within the ellipse. We also show that the optimal test for this problem is achieved by a linear projection test that is based on an explicit lower-dimensional projection of the observation vector.