arXiv 1104.45132011-04math.PR

随机矩阵之和所有特征值的尾界

Tail bounds for all eigenvalues of a sum of random matrices

Alex Gittens, Joel A. Tropp

备注：: 20 pages, 1 figure, see also arXiv:1004.4389v6

摘要：

这项工作介绍了极大极小拉普拉斯变换方法，基于累积量的矩阵拉普拉斯变换方法中开发的修改“User-friendly tail bounds for sums of random matrices”（arXiv：1004.4389v6），产生的随机自伴随矩阵的总和的每个特征值的上限和下界。这台机器是用来获得经典的Bennett，Bennett和Bernstein界的特征值类似物。两个例子证明了极大极小拉普拉斯变换的有效性。第一个问题是列稀疏化对具有正交行的矩阵的谱的影响。这里，奇异值的行为可以用类相干量来描述。第二个例子解决了随机过程的协方差阵的特征值估计的相对精度问题。标准结果的收敛性样本协方差阵提供了界限上的样本个数需要获得相对准确的谱范数，但这些结果只能保证相对准确的估计的最大特征值。最小最大拉普拉斯变换论证表明，如果最低特征值衰减足够快，在（K^2*r*log（p））/eps^2样本的数量级上，其中K是C的最佳秩-r近似的条件数，则足以确保N（0，C）随机向量的协方差阵的主要r特征值以高概率估计在1+-eps的因子内。

英文摘要：

This work introduces the minimax Laplace transform method, a modification of the cumulant-based matrix Laplace transform method developed in "User-friendly tail bounds for sums of random matrices" (arXiv:1004.4389v6) that yields both upper and lower bounds on each eigenvalue of a sum of random self-adjoint matrices. This machinery is used to derive eigenvalue analogues of the classical Chernoff, Bennett, and Bernstein bounds. Two examples demonstrate the efficacy of the minimax Laplace transform. The first concerns the effects of column sparsification on the spectrum of a matrix with orthonormal rows. Here, the behavior of the singular values can be described in terms of coherence-like quantities. The second example addresses the question of relative accuracy in the estimation of eigenvalues of the covariance matrix of a random process. Standard results on the convergence of sample covariance matrices provide bounds on the number of samples needed to obtain relative accuracy in the spectral norm, but these results only guarantee relative accuracy in the estimate of the maximum eigenvalue. The minimax Laplace transform argument establishes that if the lowest eigenvalues decay sufficiently fast, on the order of (K^2*r*log(p))/eps^2 samples, where K is the condition number of an optimal rank-r approximation to C, are sufficient to ensure that the dominant r eigenvalues of the covariance matrix of a N(0, C) random vector are estimated to within a factor of 1+-eps with high probability.

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