Concentration inequalities and moment bounds for sample covariance operators
Concentration inequalities and moment bounds for sample covariance operators
复制标题
DOI：
10.3150/15-bej730
复制
发表时间：
2017-02-01
期刊：
影响因子：
1.5
通讯作者：
Lounici, Karim
中科院分区：
文献类型：
--
作者：
Koltchinskii, Vladimir;Lounici, Karim
设X，X-1，.，X-n，.身份证可分Banach空间E中的中心高斯随机变量，其协方差算子为Sigma：Sigma：E* -> E，Sigma u = E（X，u）X，u是E* 中的元素.样本协方差算子Sigma：E* -> E定义为Sigma u：= n（-1）Sigma（n）（j=1）（X-j，U）X-j，u是E* 中的元素.特别地，证明了E平行于Sigma -Sigma平行于Sigma平行于（root r（Sigma）/n v r（Sigma）/n），其中（Sigma）：=（E）平行于X平行于）（2）/平行于Sigma -Sigma平行于- M竖线小于或近似于平行于Sigma平行于（root t/n v t/n），其中M是中位数，或平行于Sigma -Sigma平行于的期望值。另一方面，在r（Sigma）> n的假设下，对于所有t > 1，概率至少为1- e（-t.）平行于Sigma -Sigma平行于- M竖线小于或类似于平行于Sigma平行于（root r（Sigma）/n root t/n v t/n）。
Let X, X-1,..., X-n,. be i.i.d. centered Gaussian random variables in a separable Banach space E with covariance operator Sigma:Sigma:E* -> E, Sigma u = E(X,u)X, u is an element of E*.The sample covariance operator Sigma : E* -> E is defined asSigma u := n(-1) Sigma(n)(j=1) (X-j,U)X-j, u is an element of E*The goal of the paper is to obtain concentration inequalities and expectation bounds for the operator norm parallel to Sigma -Sigma parallel to of the deviation of the sample covariance operator from the true covariance operator. In particular, it is shown thatE parallel to Sigma -Sigma parallel to parallel to Sigma parallel to (root r(Sigma)/n v r(Sigma) /n),wherer(Sigma) := (E)parallel to X parallel to)(2)/parallel to Sigma parallel toparallel to Sigma -Sigma parallel to - M vertical bar less than or similar to parallel to Sigma parallel to (root t/n v t/n),where M is either the median, or the expectation of parallel to Sigma -Sigma parallel to.On the other hand, under the assumption that r(Sigma) > n, for all t > 1, with probability at least 1- e(-t.)parallel to Sigma -Sigma parallel to - M vertical bar less than or similar to parallel to Sigma parallel to (root r(Sigma)/n root t/n v t/n).