广义样本协方差矩阵极值特征值的大偏差
Large deviations of extreme eigenvalues of generalized sample covariance matrices
摘要：
本文提出了一种计算随机矩阵的最大特征值为无穷大的罕见事件概率的分析方法。其大偏差的右尾）。结果也转移到了左尾最小特征值的大偏差上。该技术改进了过去的方法，不需要明确的法律的特征值，我们将其应用到一个大类的随机矩阵，以前遥不可及。特别是，我们解决了一个开放的问题，与性能的主成分分析高度相关的数据，并打开了道路，分析复杂的推理模型的高维景观。我们使用重要性抽样方法来探测我们的结果，有效地模拟概率小至$10^{-100}$的事件。
英文摘要：
We present an analytical technique to compute the probability of rare events in which the largest eigenvalue of a random matrix is atypically large (i.e.\ the right tail of its large deviations). The results also transfer to the left tail of the large deviations of the smallest eigenvalue. The technique improves upon past methods by not requiring the explicit law of the eigenvalues, and we apply it to a large class of random matrices that were previously out of reach. In particular, we solve an open problem related to the performance of principal components analysis on highly correlated data, and open the way towards analyzing the high-dimensional landscapes of complex inference models. We probe our results using an importance sampling approach, effectively simulating events with probability as small as $10^{-100}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
