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. Comparison between formulae for the counting functions of the heights tn of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the tn are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian Hcl. Many features of Hcl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the tn have a similar structure to those of the semiclassical En; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian Hcl=XP. Let A be a complex matrix with arbitrary Jordan structure and @math an eigenvalue of A whose largest Jordan block has size n. We review previous results due to Lidskii [U.S.S. R. Comput. Math. and Math. Phys., 1 (1965), pp. 73--85], showing that the splitting of @math under a small perturbation of A of order @math is, generically, of order @math . Explicit formulas for the leading coefficients are obtained, involving the perturbation matrix and the eigenvectors of A. We also present an alternative proof of Lidskii's main theorem, based on the use of the Newton diagram. This approach clarifies certain difficulties which arise in the nongeneric case and leads, in some situations, to the extension of Lidskii's results. These results suggest a new notion of Holder condition number for multiple eigenvalues, depending only on the associated left and right eigenvectors, appropriately normalized, not on the Jordan vectors. 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. Let (a>0, b>0, ab<1; ) and let (g L^2( R ). ) In this paper we investigate the relation between the frame operator (S:f L^2( R ) n,m ,(f,g_ na,mb ) ,g_ na,mb ) and the matrix (H ) whose entries (H_ k,l ,; ,k',l' ) are given by ((g_ k' b,l' a ,g_ k b,l a ) ) for (k,l,k',l' Z . ) Here (f_ x,y (t)= exp (2 iyt) ,f(t-x), ) (t R ), for any (f L^2( R ). ) We show that (S ) is bounded as a mapping of (L^2( R ) ) into (L^2( R ) ) if and only if (H ) is bounded as a mapping of (l^2( Z ^2) ) into (l^2( Z ^2). ) Also we show that (AI S BI ) if and only if (AI 1 ab ,H BI, ) where (I ) denotes the identity operator of (L^2( R ) ) and (l^2( Z ^2), ) respectively, and (A 0, ) (B< . ) Next, when (g ) generates a frame, we have that ((g_ k b,l a )_ k,l ) has an upper frame bound, and the minimal dual function (^ ) can be computed as (ab , k,l ,(H^ -1 )_ k,l ,; ,o,o ,g_ k b,l a . ) The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a (g ) generating a frame are inherited by (^ . ) In particular, we show that (^ S ) when (g S ) generates a frame (( S ) Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula. When a Hermitian linear operator is slightly perturbed, by how much can its invariant subspaces change? Given some approximations to a cluster of neighboring eigenvalues and to the corresponding eigenvectors of a real symmetric matrix, and given an estimate for the gap that separates the cluster from all other eigenvalues, how much can the subspace spanned by the eigenvectors differ from the subspace spanned by our approximations? These questions are closely related; both are investigated here. The difference between the two subspaces is characterized in terms of certain angles through which one subspace must be rotated in order most directly to reach the other. These angles unify the treatment of natural geometric, operator-theoretic and error-analytic questions concerning those subspaces. Sharp bounds upon trigonometric functions of these angles are obtained from the gap and from bounds upon either the perturbation (1st question) or a computable residual (2nd question). An example is included. Let @math be a primitive Hilbert modular form of parallel weight @math and level @math for the totally real field @math , and let @math be a rational prime coprime to @math . If @math is ordinary at @math and @math is a CM extension of @math of relative discriminant @math prime to @math , we give an explicit construction of the @math -adic Rankin-Selberg @math -function @math . When the sign of its functional equation is @math , we show, under the assumption that all primes @math are principal ideals of @math which split in @math , that its central derivative is given by the @math -adic height of a Heegner point on the abelian variety @math associated with @math . This @math -adic Gross--Zagier formula generalises the result obtained by Perrin-Riou when @math and @math satisfies the so-called Heegner condition. We deduce applications to both the @math -adic and the classical Birch and Swinnerton-Dyer conjectures for @math . We continue the study of the Hermitian random matrix ensemble with external source Open image in new window where A has two distinct eigenvalues ±a of equal multiplicity. This model exhibits a phase transition for the value a=1, since the eigenvalues of M accumulate on two intervals for a>1, and on one interval for 0 1 was treated in Part I, where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices, that is, limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum, and in terms of the Airy kernel at the edge. In this paper we establish the same results for the case 0<a<1. As in Part I we apply the Deift Zhou steepest descent analysis to a 3×3-matrix Riemann-Hilbert problem. Due to the different structure of an underlying Riemann surface, the analysis includes an additional step involving a global opening of lenses, which is a new phenomenon in the steepest descent analysis of Riemann-Hilbert problems. Abstract Let X 1 , … , X N ∈ R n be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability one has sup x ∈ S n − 1 | 1 N ∑ i = 1 N ( | 〈 X i , x 〉 | 2 − E | 〈 X i , x 〉 | 2 ) | ⩽ C n N , where C is an absolute positive constant. This result is valid in a more general framework when the linear forms ( 〈 X i , x 〉 ) i ⩽ N , x ∈ S n − 1 and the Euclidean norms ( | X i | n ) i ⩽ N exhibit uniformly a sub-exponential decay. As a consequence, if A denotes the random matrix with columns ( X i ) , then with overwhelming probability, the extremal singular values λ min and λ max of A A ⊤ satisfy the inequalities 1 − C n N ⩽ λ min N ⩽ λ max N ⩽ 1 + C n N which is a quantitative version of Bai–Yin theorem (Z.D. Bai, Y.Q. Yin, 1993 [4] ) known for random matrices with i.i.d. entries. Given @math noisy samples with @math dimensions, where @math , we show that the multi-step thresholding procedure based on the Lasso -- we call it the Thresholded Lasso , can accurately estimate a sparse vector @math in a linear model @math , where @math is a design matrix normalized to have column @math norm @math , and @math . We show that under the restricted eigenvalue (RE) condition (Bickel-Ritov-Tsybakov 09), it is possible to achieve the @math loss within a logarithmic factor of the ideal mean square error one would achieve with an oracle while selecting a sufficiently sparse model -- hence achieving sparse oracle inequalities ; the oracle would supply perfect information about which coordinates are non-zero and which are above the noise level. In some sense, the Thresholded Lasso recovers the choices that would have been made by the @math penalized least squares estimators, in that it selects a sufficiently sparse model without sacrificing the accuracy in estimating @math and in predicting @math . We also show for the Gauss-Dantzig selector (Cand e s-Tao 07), if @math obeys a uniform uncertainty principle and if the true parameter is sufficiently sparse, one will achieve the sparse oracle inequalities as above, while allowing at most @math irrelevant variables in the model in the worst case, where @math is the smallest integer such that for @math , @math . Our simulation results on the Thresholded Lasso match our theoretical analysis excellently. Let @math be a compact Riemannian submanifold of @math of dimension @math and let @math be a sample of i.i.d. points in @math with uniform distribution. We study the random operators @math where @math is the Gaussian kernel and @math as @math Such operators can be viewed as graph laplacians (for a weighted graph with vertices at data points) and they have been used in the machine learning literature to approximate the Laplace-Beltrami operator of @math @math (divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations @math for smooth functions @math both pointwise and uniformly in @math and @math (here @math and @math is the Riemannian volume measure). In particular, we show that for any class @math of three times differentiable functions on @math with uniformly bounded derivatives @math as soon as @math and also prove asymptotic normality of @math (functional CLT) for a fixed @math and uniformly in @math Calculating portions of eigenvalues and eigenvectors of matrices or matrix pencils has many applications. An approach to this calculation for Hermitian problems based on a density matrix has been proposed in 2009 and a software package called FEAST has been developed. The density-matrix approach allows FEAST's implementation to exploit a key strength of modern computer architectures, namely, multiple levels of parallelism. Consequently, the software package has been well received and subsequently commercialized. A detailed theoretical analysis of Hermitian FEAST has also been established very recently. This paper generalizes the FEAST algorithm and theory, for the first time, to tackle non-Hermitian problems. Fundamentally, the new algorithm is basic subspace iteration or Bauer bi-iteration, except applied with a novel accelerator based on Cauchy integrals. The resulting algorithm retains the multi-level parallelism of Hermitian FEAST, making it a valuable new tool for large-scale computational science and engineering problems on leading-edge computing platforms. We investigate the clustering performances of the relaxed @math means in the setting of sub-Gaussian Mixture Model (sGMM) and Stochastic Block Model (SBM). After identifying the appropriate signal-to-noise ratio (SNR), we prove that the misclassification error decay exponentially fast with respect to this SNR. These partial recovery bounds for the relaxed @math means improve upon results currently known in the sGMM setting. In the SBM setting, applying the relaxed @math means SDP allows to handle general connection probabilities whereas other SDPs investigated in the literature are restricted to the assortative case (where within group probabilities are larger than between group probabilities). Again, this partial recovery bound complements the state-of-the-art results. All together, these results put forward the versatility of the relaxed @math means. We analyze theoretically the subspace best approximating images of a convex Lambertian object taken from the same viewpoint, but under different distant illumination conditions. We analytically construct the principal component analysis for images of a convex Lambertian object, explicitly taking attached shadows into account, and find the principal eigenmodes and eigenvalues with respect to lighting variability. Our analysis makes use of an analytic formula for the irradiance in terms of spherical-harmonic coefficients of the illumination and shows, under appropriate assumptions, that the principal components or eigenvectors are identical to the spherical harmonic basis functions evaluated at the surface normal vectors. Our main contribution is in extending these results to the single-viewpoint case, showing how the principal eigenmodes and eigenvalues are affected when only a limited subset (the upper hemisphere) of normals is available and the spherical harmonics are no longer orthonormal over the restricted domain. Our results are very close, both qualitatively and quantitatively, to previous empirical observations and represent the first essentially complete theoretical explanation of these observations. Let f 1 , ldots, f s be polynomials in Q[X 1 , ..., X n ] that generate a radical ideal and let V be their complex zero-set. Suppose that V is smooth and equidimensional; then we show that computing suitable sections of the polar varieties associated to generic projections of V gives at least one point in each connected component of V ∩ Rn. We deduce an algorithm that extends that of Bank, Giusti, Heintz and Mbakop to non-compact situations. Its arithmetic complexity is polynomial in the complexity of evaluation of the input system, an intrinsic algebraic quantity and a combinatorial quantity. Let B i be deterministic real symmetric m × m matrices, and ξ i be independent random scalars with zero mean and “of order of one” (e.g., @math ). We are interested to know under what conditions “typical norm” of the random matrix @math is of order of 1. An evident necessary condition is @math , which, essentially, translates to @math ; a natural conjecture is that the latter condition is sufficient as well. In the paper, we prove a relaxed version of this conjecture, specifically, that under the above condition the typical norm of S N is @math : @math for all Ω > 0 We outline some applications of this result, primarily in investigating the quality of semidefinite relaxations of a general quadratic optimization problem with orthogonality constraints @math , where F is quadratic in X = (X 1,... ,X k ). We show that when F is convex in every one of X j , a natural semidefinite relaxation of the problem is tight within a factor slowly growing with the size m of the matrices @math . A stronger result on the limiting distribution of the eigenvalues of random Hermitian matrices of the form A + XTX*, originally studied in Marcenko and Pastur, is presented. Here, X(N - n), T(n - n), and A(N - N) are independent, with X containing i.i.d. entries having finite second moments, T is diagonal with real (diagonal) entries, A is Hermitian, and n N --> c > 0 as N --> [infinity]. Under additional assumptions on the eigenvalues of A and T, almost sure convergence of the empirical distribution function of the eigenvalues of A + XTX* is proven with the aid of Stieltjes transforms, taking a more direct approach than previous methods.