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 $10^{-100}$.

翻译：我们提出一种分析技术,以计算随机矩阵的最大电子元值异常大(即其大偏差的右尾巴)的罕见事件的概率;结果也转移到最小电子元值大偏差的左尾部; 技术在以往方法上有所改进,不需要明确的电子元值定律, 我们将其应用到以前无法接触到的大量随机矩阵中。 特别是, 我们解决了一个与对高度相关数据进行主要组成部分分析有关的未决问题, 并为分析复杂推断模型的高维景观开辟了道路。 我们使用重要取样方法检测我们的结果, 有效地模拟概率小于10美元- 100美元的事件。