We prove a Saad's type bound for harmonic Ritz vectors of a Hermitian matrix. The new bound reveals a dependence of the harmonic Rayleigh--Ritz procedure on the condition number of a shifted problem operator. Several practical implications are discussed. In particular, the bound motivates incorporation of preconditioning into the harmonic Rayleigh--Ritz scheme.
In particular, if @math is Hermitian, the result of @cite_4 states that eq:chen.jia (x, v) 1 + ^2 |B^ -1 |^2 sep ( ,G)^2 (x, K), where @math , @math , and @math denotes an orthogonal projector onto @math . The quantity @math describes separation of the targeted eigenvalue @math from the harmonic Ritz values different from the one associated with @math , which are the eigenvalues of @math ; see @cite_4 for the precise definition of @math .
@cite_4
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Let (λ,x) be an eigenpair of the matrix A of order n and let (µ,u) be a Ritz pair of A with respect to a subspace K. Saad has derived a simple priori error bound for sin ∠(x, u) in terms of sin ∠(x, K) for A Hermitian. Similar to Saad's result, Stewart has got an equally simple inequality for A non-Hermitian. In this paper, let (θ, w) be a harmonic Ritz pair from a subspace K, a similar priori error bound for sin ∠(x, w) is established in terms of sin ∠(x, K). Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090. We consider @math Hermitian matrices with i.i.d. entries @math whose tail probabilities @math behave like @math for some @math and @math . We establish a large deviation principle for the empirical spectral measure of @math with speed @math with a good rate function @math that is finite only if @math is of the form @math for some probability measure @math on @math , where @math denotes the free convolution and @math is Wigner's semicircle law. We obtain explicit expressions for @math in terms of the @math th moment of @math . The proof is based on the analysis of large deviations for the empirical distribution of very sparse random rooted networks. 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.