This work introduces the minimax Laplace transform method, a modification of the cumulant-based matrix Laplace transform method developed in [Tro11c] that yields bothudupper and lower bounds on each eigenvalue of a sum of random self-adjoint matrices. This machinery is usedudto derive eigenvalue analogs of the classical Chernoff, Bennett, and Bernstein bounds.ududTwo 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 secondudexample addresses the question of relative accuracy in the estimation of eigenvalues of the covarianceudmatrix of a random process. Standard results on the convergence of sample covariance matricesudprovide 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, Ω(ε^(-2)κ^2_ℓ ℓ log p) samples, where κ_ℓ = λ_1(C)/λ_ℓ(C), are sufficient to ensure that the dominant ℓ eigenvalues of the covariance matrix of a N(0,C) random vector are estimated to within a factor of 1 ± ε with high probability.
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机译：这项工作介绍了minimax拉普拉斯变换方法，该模型是对[Tro11c]中开发的基于累积量的矩阵拉普拉斯变换方法的修改，该方法在随机自伴随矩阵之和的每个特征值上产生 upupper和下界。该工具用于推导经典的Chernoff，Bennett和Bernstein边界的特征值类似物。 ud ud两个示例演示了minimax Laplace变换的功效。第一个问题涉及列稀疏化对具有正交行的矩阵的光谱的影响。在这里，奇异值的行为可以用相干量来描述。第二个 udexample解决了随机过程协方差 udmatrix的特征值估计中的相对准确性问题。样本协方差矩阵收敛的标准结果获得频谱范数中相对精度所需的样本数量的边界不足，但是这些结果仅保证最大特征值的估计中的相对精度。最小极大拉普拉斯变换论点确定，如果最低特征值衰减足够快，则Ω（ε^（-2）κ^2_ℓlog p）样本足以确保满足以下条件：κ_ℓ=λ_1（C）/λ_ℓ（C） N（0，C）随机向量协方差矩阵的优势特征值估计概率在1±ε范围内。
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