Perturbation theory for normal operators

Rainer, Armin

doi:10.1090/s0002-9947-2013-05854-0

Cited by 16 publications

(15 citation statements)

References 38 publications

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“…(3) If A is C 2 then there exists a twice differentiable parameterization of the eigenvalues. The first result follows from a result due to Weyl [39], the second and third were shown in [28]. Actually, these results are true for normal complex matrices and, in appropriate form, even for normal operators in Hilbert space with common domain of definition and compact resolvents; see [28].…”

Section: Further Examplesmentioning

confidence: 73%

Lifting Differentiable Curves From Orbit Spaces

Parusiński

Rainer

2015

Transformation Groups

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Let ρ : G → O(V ) be a real finite dimensional orthogonal representation of a compact Lie group, let σ = (σ 1 , . . . , σ n ) : V → R n , where σ 1 , . . . , σ n form a minimal system of homogeneous generators of the G-invariant polynomials on V , and set d = max i deg σ i . We prove that for each C d−1,1 -curve c in σ(V ) ⊆ R n there exits a locally Lipschitz lift over σ, i.e., a locally Lipschitz curve c in V so that c = σ • c, and we obtain explicit bounds for the Lipschitz constant of c in terms of c. Moreover, we show that each C d -curve in σ(V ) admits a C 1 -lift. For finite groups G we deduce a multivariable version and some further results.

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Section: Further Examplesmentioning

confidence: 73%

Lifting Differentiable Curves From Orbit Spaces

Parusiński

Rainer

2015

Transformation Groups

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“…Ideally, we would have wanted to use the proof of Lemma 4.5 and state that since the matrix R is smooth in r and q then there exists a smooth choice of RR T 's eigenvectors (i.e., the basis {u k } of H ). In the general case of smooth multivariate perturbations of matrices, this is not always true (e.g., see [31,17]). Nevertheless, in the following lemma, we are able to show that in our case the projections onto H (r; q) vary smoothly in both parameters.…”

Section: Smoothness Of the Coordinate System Hmentioning

confidence: 99%

Manifold Approximation by Moving Least-Squares Projection (MMLS)

Sober

Levin

2019

Constr Approx

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In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes referred to as manifold learning) assume that the scattered input data is lying on a lower dimensional manifold, thus the high dimensionality problem can be overcome by learning the lower dimensionality behavior. However, in real life applications, data is often very noisy. In this work, we propose a method to approximate M a d-dimensional C m+1 smooth submanifold of R n (d << n) based upon noisy scattered data points (i.e., a data cloud). We assume that the data points are located "near" the lower dimensional manifold and suggest a non-linear moving least-squares projection on an approximating d-dimensional manifold. Under some mild assumptions, the resulting approximant is shown to be infinitely smooth and of high approximation order (i.e., O(h m+1 ), where h is the fill distance and m is the degree of the local polynomial approximation). The method presented here assumes no analytic knowledge of the approximated manifold and the approximation algorithm is linear in the large dimension n. Furthermore, the approximating manifold can serve as a framework to perform operations directly on the high dimensional data in a computationally efficient manner. This way, the preparatory step of dimension reduction, which induces distortions to the data, can be avoided altogether.

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“…Without normality even real analytic curves of diagonalisable matrices need not admit smooth choices of the eigenvalues. All this can be found in [33] and the references therein. For the optimal (Sobolev) regularity of the eigenvalues of smooth curves of arbitrary quadratic matrices see [32].…”

Section: Rotating Solutionsmentioning

confidence: 99%