We show that, under mild assumptions, the spectrum of a sum of independent random matrices is close to that of the Gaussian random matrix whose entries have the same mean and covariance. This nonasymptotic universality principle yields sharp matrix concentration inequalities for general sums of independent random matrices when combined with the Gaussian theory of Bandeira, Boedihardjo, and Van Handel. A key feature of the resulting theory is that it is applicable to a broad class of random matrix models that may have highly nonhomogeneous and dependent entries, which can be far outside the mean-field situation considered in classical random matrix theory. We illustrate the theory in applications to random graphs, matrix concentration inequalities for smallest singular values, sample covariance matrices, strong asymptotic freeness, and phase transitions in spiked models.

我们表明，在温和的假设下，独立随机矩阵之和的频谱接近于高斯随机矩阵的频谱，其条目具有相同的均值和协方差。当与 Bandeira、Boedihardjo 和 Van Handel 的高斯理论相结合时，这种非渐近普遍性原理会产生独立随机矩阵的一般和的尖锐矩阵浓度不等式。所得理论的一个关键特征是，它适用于一大类随机矩阵模型，这些模型可能具有高度非齐次和依赖的条目，这可能远远超出经典随机矩阵理论中考虑的平均场情况。我们说明了该理论在随机图中的应用、最小奇异值的矩阵浓度不等式、样本协方差矩阵、强渐近自由度和加标模型中的相变。