Rayleigh-Schr " o dinger perturbation theory is a well-known theory in quantum mechanics and it offers useful characterization of eigenvectors of a perturbed matrix. Suppose @math and perturbation @math are both Hermitian matrices, @math , @math are eigenvalues of @math in descending order, and @math are leading eigenvectors of @math and @math . Rayleigh-Schr " o dinger theory shows asymptotically, @math where @math . However, the asymptotic theory does not apply to larger @math ; in particular, it fails when @math . In this paper, we present a nonasymptotic theory with @math being a random matrix. We prove that, when @math and @math has independent and centered subgaussian entries above its diagonal, with high probability, for all @math simultaneously, under a condition on eigenvalues of @math that involves all gaps @math . This bound is valid, even in cases where @math . The result is optimal, except for a log term. It also leads to an improvement of Davis-Kahan theorem.
In classical paper by @cite_5 , only deterministic matrices were considered. @cite_6 established similar bounds for singular value decomposition, which is similar to spectral decomposition but applies to non-Hermitian matrices as well. Recently, @cite_11 studied @math perturbation bounds for structured low-rank matrices that have incoherent eigenvectors, but their result is suboptimal when perturbation matrix is random.
@cite_5 @cite_6 @cite_11
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We describe two recently proposed randomized algorithms for the construction of low-rank approximations to matrices, and demonstrate their application (inter alia) to the evaluation of the singular value decompositions of numerically low-rank matrices. Being probabilistic, the schemes described here have a finite probability of failure; in most cases, this probability is rather negligible (10−17 is a typical value). In many situations, the new procedures are considerably more efficient and reliable than the classical (deterministic) ones; they also parallelize naturally. We present several numerical examples to illustrate the performance of the schemes. We compute the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. We consider random Hermitian matrices with independent Gaussian entries Mij,i≤j with various expectations. We prove that the largest eigenvalue of such random matrices exhibits, in the large N limit, various limiting distributions depending on both the eigenvalues of the matrix Open image in new window and its rank. This rank is also allowed to increase with N in some restricted way. This paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix. These results improve on prior work by Candes and Recht, Candes and Tao, and Keshavan, Montanari, and Oh. The reconstruction is accomplished by minimizing the nuclear norm, or sum of the singular values, of the hidden matrix subject to agreement with the provided entries. If the underlying matrix satisfies a certain incoherence condition, then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition. The proof of this assertion is short, self contained, and uses very elementary analysis. The novel techniques herein are based on recent work in quantum information theory. Given a matrix A, it is often desirable to find a good approximation to A that has low rank. We introduce a simple technique for accelerating the computation of such approximations when A has strong spectral features, that is, when the singular values of interest are significantly greater than those of a random matrix with size and entries similar to A. Our technique amounts to independently sampling and or quantizing the entries of A, thus speeding up computation by reducing the number of nonzero entries and or the length of their representation. Our analysis is based on observing that the acts of sampling and quantization can be viewed as adding a random matrix N to A, whose entries are independent random variables with zero-mean and bounded variance. Since, with high probability, N has very weak spectral features, we can prove that the effect of sampling and quantization nearly vanishes when a low-rank approximation to A p N is computed. We give high probability bounds on the quality of our approximation both in the Frobenius and the 2-norm.