Bootstrapping the Error of Oja's Algorithm
Bootstrapping the Error of Oja's Algorithm
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发表时间：
2021-06
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通讯作者：
Robert Lunde;Purnamrita Sarkar;Rachel A. Ward
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作者：
Robert Lunde;Purnamrita Sarkar;Rachel A. Ward
我们考虑了 Oja 流式主成分分析算法中主特征向量估计误差的量化不确定性问题，其中数据是根据某些未知分布生成的 IID。通过将 U 统计文献中的经典工具与随机向量二次形式和矩阵乘积集中的高维中心极限定理的最新结果相结合，我们为总体特征向量与 Oja 算法输出之间的 $\sin^2$ 误差建立了加权 $\chi^2$ 近似结果。由于估计与近似分布相关的协方差矩阵需要了解未知的模型参数，因此我们提出了一种可以在线更新的乘法器引导算法。我们建立了引导分布以高概率接近相应采样分布的条件，从而将引导分布建立为适当渐近状态下的一致推理方法。
We consider the problem of quantifying uncertainty for the estimation error of the leading eigenvector from Oja's algorithm for streaming principal component analysis, where the data are generated IID from some unknown distribution. By combining classical tools from the U-statistics literature with recent results on high-dimensional central limit theorems for quadratic forms of random vectors and concentration of matrix products, we establish a weighted $\chi^2$ approximation result for the $\sin^2$ error between the population eigenvector and the output of Oja's algorithm. Since estimating the covariance matrix associated with the approximating distribution requires knowledge of unknown model parameters, we propose a multiplier bootstrap algorithm that may be updated in an online manner. We establish conditions under which the bootstrap distribution is close to the corresponding sampling distribution with high probability, thereby establishing the bootstrap as a consistent inferential method in an appropriate asymptotic regime.