论文标题
广义样品协方差矩阵的极端特征值的大偏差
Large deviations of extreme eigenvalues of generalized sample covariance matrices
论文作者
论文摘要
我们提出了一种分析技术，以计算罕见事件的概率，其中随机基质的最大特征值在其大偏差的右尾巴上是非典型的。结果还传递到最小特征值大偏差的左尾。该技术通过不需要特征值的明确定律而改善了过去的方法，并且我们将其应用于以前无法实现的大量随机矩阵。特别是，我们解决了与高度相关数据的主要组件分析的性能有关的开放问题，并为分析复杂推理模型的高维景观开辟了道路。我们使用重要性采样方法探测结果，有效地模拟了概率至$ 10^{ - 100} $的事件。
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 $10^{-100}$.
