Unitarily Invariant Matrix Norm Research Articles

Tikhonov regularization with MTRSVD method for solving large-scale discrete ill-posed problems

Regularization is possibly the most popular method for solving discrete ill-posed problems, whose solution is less sensitive to the error in the observed vector in the right hand than the original solution. This paper presents a new modified truncated randomized singular value decomposition (TR-MTRSVD) method for large Tikhonov regularization in standard form. The proposed TR-MTRSVD algorithm introduces the idea of randomized algorithm into the improved truncated singular value decomposition (MTSVD) method to solve large Tikhonov regularization problems. The approximation matrix Ãℓ produced by randomized SVD is replaced by the closest matrix Ãk̃ in a unitarily invariant matrix norm with the same spectral condition number. The regularization parameters are determined by the discrepancy principle. Numerical examples show the effectiveness and efficiency of the proposed TR-MTRSVD algorithm for large Tikhonov regularization problems.

When does ‖f(A)‖ = f(‖A‖) hold true?

Let a strongly stable norm on the set of complex n-by-n matrices be given, which means that for all and all . Furthermore, let be a power series with nonnegative coefficients and radius of convergence R>0. If , we additionally suppose that . We aim to characterize those A with , which fulfil . We first show how to reduce the discussion of f to Neumann series. For matrix norms induced by uniformly convex vector norms, like the -norms, , it follows from known results of the Daugavet equation that holds true if, and only if, is an eigenvalue of A, provided that for some . Under adapted assumptions on the we prove that this equivalence remains true for the - and the -norm, for unitarily invariant matrix norms and for the numerical radius. We conjecture this equivalence to be valid for all strongly stable norms if for all .

A novel modified TRSVD method for large-scale linear discrete ill-posed problems

The truncated singular value decomposition (TSVD) is a popular method for solving linear discrete ill-posed problems with a small to moderately sized matrix A. This method replaces the matrix A by the closest matrix Ak of low rank k, and then computes the minimal norm solution of the linear system of equations with a rank-deficient matrix so obtained. The modified TSVD (MTSVD) method improves the TSVD method, by replacing A by a matrix that is closer to A than Ak in a unitarily invariant matrix norm and has the same spectral condition number as Ak. Approximations of the SVD of a large matrix A can be computed quite efficiently by using a randomized SVD (RSVD) method. This paper presents a novel modified truncated randomized singular value decomposition (MTRSVD) method for computing approximate solutions to large-scale linear discrete ill-posed problems. The rank, k, is determined with the aid of the discrepancy principle, but other techniques for selecting a suitable rank also can be used. Numerical examples illustrate the effectiveness of the proposed method and compare it to the truncated RSVD method.

A note on unitarily invariant matrix norms

Given a unitarily invariant matrix norm ‖⋅‖ on n×n (real or complex) matrices, characterization is given for matrices A and B such that ‖AB‖=‖A‖‖B‖.

Graph rigidity for unitarily invariant matrix norms

A rigidity theory is developed for bar-joint frameworks in linear matrix spaces endowed with a unitarily invariant matrix norm. Analogues of Maxwell's counting criteria are obtained and minimally rigid matrix frameworks are shown to belong to the matroidal class of (k,l)-sparse graphs for suitable k and l. An edge-colouring technique is developed to characterise infinitesimal rigidity for product norms and then applied to show that the graph of a minimally rigid bar-joint framework in the space of 2×2 symmetric (respectively, hermitian) matrices with the trace norm admits an edge-disjoint packing consisting of a (Euclidean) rigid graph and a spanning tree.

Unitarily invariant errors-in-variables estimation

Linear relations, containing measurement errors in the input and output data, are considered. Parameters of these errors-in-variables (EIV) models can be estimated by minimizing the total least squares (TLS) of the input–output disturbances, i.e., penalizing the orthogonal squared misfit. This approach corresponds to minimizing the Frobenius norm of the error matrix. An extension of the traditional TLS estimator in the EIV model—the EIV estimator—is proposed in the way that a general unitarily invariant norm of the error matrix is minimized. Such an estimator is highly non-linear. Regardless of the chosen unitarily invariant matrix norm, the corresponding EIV estimator is shown to coincide with the TLS estimator. Its existence and uniqueness is discussed. Moreover, the EIV estimator is proved to be scale invariant, interchange, direction, and rotation equivariant.

A modified truncated singular value decomposition method for discrete ill-posed problems

SUMMARY Truncated singular value decomposition is a popular method for solving linear discrete ill-posed problems with a small to moderately sized matrix A. Regularization is achieved by replacing the matrix A by its best rank-k approximant, which we denote by Ak. The rank may be determined in a variety of ways, for example, by the discrepancy principle or the L-curve criterion. This paper describes a novel regularization approach, in which A is replaced by the closest matrix in a unitarily invariant matrix norm with the same spectral condition number as Ak. Computed examples illustrate that this regularization approach often yields approximate solutions of higher quality than the replacement of A by Ak.Copyright © 2014 John Wiley & Sons, Ltd.

Low-Rank Positive Approximants of Symmetric Matrices

Given a symmetric matrix X, we consider the problem of finding a low-rank positive approximant of X. That is, a symmetric positive semidefinite matrix, S, whose rank is smaller than a given positive integer, , which is nearest to X in a certain matrix norm. The problem is first solved with regard to four common norms: The Frobenius norm, the Schatten p-norm, the trace norm, and the spectral norm. Then the solution is extended to any unitarily invariant matrix norm. The proof is based on a subtle combination of Ky Fan dominance theorem, a modified pinching principle, and Mirsky minimum-norm theorem.

On vector and matrix median computation

The aim of this paper is to gain more insight into vector and matrix medians and to investigate algorithms to compute them. We prove relations between vector and matrix means and medians, particularly regarding the classical structure tensor. Moreover, we examine matrix medians corresponding to different unitarily invariant matrix norms for the case of symmetric 2×2 matrices, which frequently arise in image processing. Our findings are explained and illustrated by numerical examples. To solve the corresponding minimization problems, we propose several algorithms. Existing approaches include Weiszfeld’s algorithm for the computation of ℓ2 vector medians and semi-definite programming, in particular, second order cone programming, which has been used for matrix median computation. In this paper, we adapt Weiszfeld’s algorithm for our setting and show that also two splitting methods, namely the alternating direction method of multipliers and the parallel proximal algorithm, can be applied for generalized vector and matrix median computations. Besides, we compare the performance of these algorithms numerically and apply them within local median filters.

Condition numbers and D-efficiency

Suppose that experimental designs have comparable A-efficiencies for estimation, the choice among designs then depending on their comparative D-efficiencies. Using the theory of majorization, it is shown that designs having greater D-efficiencies are more ill-conditioned, using condition numbers based on the unitarily invariant matrix norms. Implications are drawn regarding ridge regression, Mauchly (1940) sphericity criterion, and the structure of joint confidence regions for principal functions of the parameters. A case study treats alternative designs for estimating parameters of a second-order response model.

Mesh Movement and Metamorphosis

Mesh coarsening and mesh enrichment are combined with a r-refinement scheme to produce a flexible approach for mesh adaptation of time evolving domains. The robustness of this method depends heavily upon maintaining mesh quality during each adaptation cycle. This in turn is inlfuenced by the ability to identify and remove badly shaped elements after the r-refinement stage. Measures of both element quality and element deformation can be defined in terms of unitarily invariant matrix norms. The construction of these element deformation and quality measures is described, and details are provided of the three stages of the adaptation cycle.

Perturbation Theory for Simultaneous Bases of Singular Subspaces

New perturbation theorems are proved for simultaneous bases of singular subspaces of real matrices. These results improve the absolute bounds previously obtained in [6] for general (complex) matrices. Unlike previous results, which are valid only for the Frobenius norm, the new bounds, as well as those in [6] for complex matrices, are extended to any unitarily invariant matrix norm. The bounds are complemented with numerical experiments which show their relevance for the algorithms computing the singular value decomposition. Additionally, the differential calculus approach employed allows to easily prove new sin Θ perturbation theorems for singular subspaces which deal independently with left and right singular subspaces.

Convex analysis on Cartan subspaces

In 1937, von Neumann [31] gave a famous characterization of unitarily invariant matrix norms (that is, norms f on Cp×q satisfying f(uxv) = f(x) for all unitary matrices u and v and matrices x in Cp×q). His result states that such norms are those functions of the form g ◦ , where the map x ∈ Cp×q 7→ (x) ∈ R has components the singular values 1(x) ≥ 2(x) ≥ · · · ≥ p(x) of x (assuming p ≤ q) and g is a norm on Rp, invariant under sign changes and permutations of components. Furthermore, he showed the respective dual norms satisfy

Structured dispersion and validity in linear inference

Structured matrices ∑(γ) = [I n + eγ′ + γe′ − \\ ̄ ggee′] arise in nonstandard linear models, where e′ = [1, …, 1], γ′ = [ γ 1, …, γ n ], and \\ ̄ gg = (γ 1 + … + γ n) n . Their properties are studied, including expressions for eigenvalues, conditions for positive definiteness, and conditioning of bE(γ) as γ varies. It is shown that if γ majorizes γ 0, then the condition numbers are ordered as c φ (∑( γ)) ⩾ c φ (∑( γ 0)) for every condition number { c φ (·); φ ∈ ϱ} generated by the unitarily invariant matrix norms. Applications are noted in linear inference and in outlier detection.

A note on the arithmetic-geometric mean inequality for every unitarily invariant matrix norm

We integrate ten unitarily invariant matrix norm inequalities equivalent to the Heinz inequality.

Minimal properties of Moore-Penrose inverses

Let Ax = y be consistent; let x 0 = Gy be any minimum-norm solution satisfying ( AG)′ = AG; and let A + be the Moore-Penrose inverse of A. It is shown that φ( G) ⩾ φ( A +) for any φ in a class Φ containing the unitarily invariant matrix norms. The conditioning of the system Ax = y is studied via condition numbers C φ( A, G). It is shown that C φ( A, G) ⩾ C φ( A, A +) for every φ∈ Φ. Moreover, bounds on C φ( A, G) are given in terms of singular values. Parallel results are found when A and G are symmetric, with applications to linear models of less than full rank.

Unitary invariance and spectral variation

We call a norm on operators or matrices weakly unitarily invariant if its value at operator A is not changed by replacing A by U∗AU, provided only that U is unitary. This class includes such norms as the numerical radius. We extend to all such norms an inequality that bounds the spectral variation when a normal operator A is replaced by another normal B in terms of the arclength of any normal path from A to B, computed using the norm in question. Related results treat the local metric geometry of the “manifold” of normal operators. We introduce a representation for weakly unitarily invariant matrix norms in terms of function norms over the unit ball, and identify this correspondence explicitly in certain cases.