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.
@cite_10 and @cite_7 studied the top singular vectors (containing eigenvectors as a special case) of a perturbed (rectangular) matrix. To compare our result with theirs, suppose the matrices under consideration are symmetric. When @math is of low-rank, and @math , and the perturbation is random, they proved upper bounds that improve over Davis-Kahan theorem (or Wedin's theorem ). Roughly speaking, they showed that when @math , with large probability, See Theorem 9 in @cite_7 . They also proved a more general theorem that requires a weaker assumption on the random model. Their bound is the same as what we obtained (see Example ), except for a log term. Yet it is not clear why their result requires @math , and how @math helps. Our nonasymptotic RS perturbation theory offers a natural explanation and generalizes their results (with the price of a stronger independence assumption).
@cite_10 @cite_7
2295296939 2951358417 1640765157 2963583445
In statistics and machine learning, people are often interested in the eigenvectors (or singular vectors) of certain matrices (e.g. covariance matrices, data matrices, etc). However, those matrices are usually perturbed by noises or statistical errors, either from random sampling or structural patterns. One usually employs Davis-Kahan @math theorem to bound the difference between the eigenvectors of a matrix @math and those of a perturbed matrix @math , in terms of @math norm. In this paper, we prove that when @math is a low-rank and incoherent matrix, the @math norm perturbation bound of singular vectors (or eigenvectors in the symmetric case) is smaller by a factor of @math or @math for left and right vectors, where @math and @math are the matrix dimensions. The power of this new perturbation result is shown in robust covariance estimation, particularly when random variables have heavy tails. There, we propose new robust covariance estimators and establish their asymptotic properties using the newly developed perturbation bound. Our theoretical results are verified through extensive numerical experiments. This text is about spiked models of non Hermitian random matrices. More specifically, we consider matrices of the type @math , where the rank of @math stays bounded as the dimension goes to infinity and where the matrix @math is a non Hermitian random matrix, satisfying an isotropy hypothesis: its distribution is invariant under the left and right actions of the unitary group. The macroscopic eigenvalue distribution of such matrices is governed by the so called Single Ring Theorem, due to Guionnet, Krishnapur and Zeitouni. We first prove that if @math has some eigenvalues out of the maximal circle of the single ring, then @math has some eigenvalues (called outliers) in the neighborhood of those of @math , which is not the case for the eigenvalues of @math in the inner cycle of the single ring. Then, we study the fluctuations of the outliers of @math around the eigenvalues of @math and prove that they are distributed as the eigenvalues of some finite dimensional random matrices. Such facts had already been noticed for Hermitian models. More surprising facts are that outliers can here have very various rates of convergence to their limits (depending on the Jordan Canonical Form of @math ) and that some correlations can appear between outliers at a macroscopic distance from each other (a fact already noticed by Knowles and Yin in the Hermitian case, but only in the case of non Gaussian models, whereas spiked Gaussian matrices belong to our model and can have such correlated outliers). Our first result generalizes a previous result by Tao for matrices with i.i.d. entries, whereas the second one (about the fluctuations) is new. Matrix perturbation inequalities, such as Weyl's theorem (concerning the singular values) and the Davis-Kahan theorem (concerning the singular vectors), play essential roles in quantitative science; in particular, these bounds have found application in data analysis as well as related areas of engineering and computer science. In many situations, the perturbation is assumed to be random, and the original matrix has certain structural properties (such as having low rank). We show that, in this scenario, classical perturbation results, such as Weyl and Davis-Kahan, can be improved significantly. We believe many of our new bounds are close to optimal and also discuss some applications. Abstract We study the recovery of Hermitian low rank matrices X ∈ C n × n from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form a j a j ⁎ for some measurement vectors a 1 , … , a m , i.e., the measurements are given by b j = tr ( X a j a j ⁎ ) . The case where the matrix X = x x ⁎ to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements b j = | 〈 x , a j 〉 | 2 ) via the PhaseLift approach, which has been introduced recently. We derive bounds for the number m of measurements that guarantee successful uniform recovery of Hermitian rank r matrices, either for the vectors a j , j = 1 , … , m , being chosen independently at random according to a standard Gaussian distribution, or a j being sampled independently from an (approximate) complex projective t-design with t = 4 . In the Gaussian case, we require m ≥ C r n measurements, while in the case of 4-designs we need m ≥ Cr n log ⁡ ( n ) . Our results are uniform in the sense that one random choice of the measurement vectors a j guarantees recovery of all rank r-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate 4-designs generalizes and improves a recent bound on phase retrieval due to Gross, Krahmer and Kueng. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.