论文标题
普遍性和尖锐的基质浓度不平等
Universality and sharp matrix concentration inequalities
论文作者
论文摘要
我们表明，在温和的假设下，独立随机矩阵总和的频谱接近于该条目具有相同均值和协方差的高斯随机矩阵的频谱。当与Bandeira，Boedihardjo和Van Handel的高斯理论结合使用时，这种非反应性普遍性原理对于独立随机矩阵的一般总和产生了急剧的基质浓度不等式。结果理论的一个关键特征是它适用于可能具有高度非均匀和依赖的条目的广泛的随机矩阵模型，这可能远远超出经典随机矩阵理论中考虑的平均场情况。我们说明了对随机图的应用，最小奇异值的矩阵浓度不等式，样品协方差矩阵，强渐近线条和峰值模型中的相变。
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.
