We study the operator norm discrepancy of i.i.d. random matrices, initiating the matrix-valued analog of a long line of work on the $\ell^{\infty}$ norm discrepancy of i.i.d. random vectors. First, we give a new analysis of the matrix hyperbolic cosine algorithm of Zouzias (2011), a matrix version of an online vector discrepancy algorithm of Spencer (1977) studied for average-case inputs by Bansal and Spencer (2020), for the case of i.i.d. random matrix inputs. We both give a general analysis and extract concrete bounds on the discrepancy achieved by this algorithm for matrices with independent entries and positive semidefinite matrices drawn from Wishart distributions. Second, using the first moment method, we give lower bounds on the discrepancy of random matrices, in particular showing that the matrix hyperbolic cosine algorithm achieves optimal discrepancy up to logarithmic terms in several cases. We both treat the special case of the Gaussian orthogonal ensemble and give a general result for low-rank matrix distributions that we apply to orthogonally invariant random projections.

翻译：我们研究了独立同分布随机矩阵的算子范数差异，开启了独立同分布随机向量$\ell^{\infty}$范数差异长期研究工作的矩阵类比。首先，我们对Zouzias (2011)提出的矩阵双曲余弦算法进行了新的分析——该算法是Spencer (1977)在线向量差异算法（Bansal和Spencer (2020)研究了其平均情形的输入）的矩阵版本，专门针对独立同分布随机矩阵输入的情形。我们既给出了通用分析，也为具有独立元素的矩阵和服从Wishart分布的正半定矩阵提取了该算法所达到差异的具体界。其次，利用一阶矩方法，我们给出了随机矩阵差异的下界，特别证明了在某些情形下，矩阵双曲余弦算法在忽略对数项的意义上达到了最优差异。我们既处理了高斯正交系综的特例，也为低秩矩阵分布推导了通用结果，并将其应用于正交不变随机投影。