The Ideal of σ-Nuclear Operators and Its Associated Tensor Norm

We introduce a new tensor norm ( σ -tensor norm) and show that it is associated with the ideal of σ -nuclear operators. In this paper, we investigate the ideal of σ -nuclear operators and the σ -tensor norm.

Approximation properties of tensor norms and operator ideals for Banach spaces

For a finitely generated tensor norm α \alpha , we investigate the α \alpha -approximation property ( α \alpha -AP) and the bounded α \alpha -approximation property (bounded α \alpha -AP) in terms of some approximation properties of operator ideals. We prove that a Banach space X has the λ \lambda -bounded α p , q {\alpha }_{p,q} -AP ( 1 ≤ p , q ≤ ∞ , 1 / p + 1 / q ≥ 1 ) (1\le p,q\le \infty ,1/p+1/q\ge 1) if it has the λ \lambda -bounded g p {g}_{p} -AP. As a consequence, it follows that if a Banach space X has the λ \lambda -bounded g p {g}_{p} -AP, then X has the λ \lambda -bounded w p {w}_{p} -AP.

Hyperspectral Image Denoising Using Global Weighted Tensor Norm Minimum and Nonlocal Low-Rank Approximation

A hyperspectral image (HSI) contains abundant spatial and spectral information, but it is always corrupted by various noises, especially Gaussian noise. Global correlation (GC) across spectral domain and nonlocal self-similarity (NSS) across spatial domain are two important characteristics for an HSI. To keep the integrity of the global structure and improve the details of the restored HSI, we propose a global and nonlocal weighted tensor norm minimum denoising method which jointly utilizes GC and NSS. The weighted multilinear rank is utilized to depict the GC information. To preserve structural information with NSS, a patch-group-based low-rank-tensor-approximation (LRTA) model is designed. The LRTA makes use of Tucker decompositions of 4D patches, which are composed of a similar 3D patch group of HSI. The alternating direction method of multipliers (ADMM) is adapted to solve the proposed models. Experimental results show that the proposed algorithm can preserve the structural information and outperforms several state-of-the-art denoising methods.

Theoretical and Experimental Analyses of Tensor-Based Regression and Classification

We theoretically and experimentally investigate tensor-based regression and classification. Our focus is regularization with various tensor norms, including the overlapped trace norm, the latent trace norm, and the scaled latent trace norm. We first give dual optimization methods using the alternating direction method of multipliers, which is computationally efficient when the number of training samples is moderate. We then theoretically derive an excess risk bound for each tensor norm and clarify their behavior. Finally, we perform extensive experiments using simulated and real data and demonstrate the superiority of tensor-based learning methods over vector- and matrix-based learning methods.

Horizontal Momentum Diffusion in GCMs Using the Dynamic Smagorinsky Model

A dynamic version of Smagorinsky’s diffusion scheme is presented that is applicable for large-eddy simulations (LES) of the atmospheric dynamics. The approach is motivated (i) by the incompatibility of conventional hyperdiffusion schemes with the conservation laws, and (ii) because the conventional Smagorinsky model (which fulfills the conservation laws) does not maintain scale invariance, which is mandatory for a correct simulation of the macroturbulent kinetic energy spectrum. The authors derive a two-dimensional (horizontal) formulation of the dynamic Smagorinsky model (DSM) and present three solutions of the so-called Germano identity: the method of least squares, a solution without invariance of the Smagorinsky parameter, and a tensor-norm solution. The applicability of the tensor-norm approach is confirmed in simulations with the Kühlungsborn mechanistic general circulation model (KMCM). The standard spectral dynamical core of the model facilitates the implementation of the test filter procedure of the DSM. Various energy spectra simulated with the DSM and the conventional Smagorinsky scheme are presented. In particular, the results show that only the DSM allows for a reasonable spectrum at all scales. Latitude–height cross sections of zonal-mean fluid variables are given and show that the DSM preserves the main features of the atmospheric dynamics. The best ratio for the test-filter scale to the resolution scale is found to be 1.33, resulting in dynamically determined Smagorinsky parameters cS from 0.10 to 0.22 in the troposphere. This result is very similar to other values of cS found in previous three-dimensional applications of the DSM.

A stress-based formulation of the free material design problem with the trace constraint and single loading condition

The problem to find an optimal distribution of elastic moduli within a given plane domain to make the compliance minimal under the condition of a prescribed value of the integral of the trace of the elastic moduli tensor is called the free material design with the trace constraint. The present paper shows that this problem can be reduced to a new problem of minimization of the integral of the stress tensor norm over stresses being statically admissible. The eigenstates and Kelvin’s moduli of the optimal Hooke tensor are determined by the stress state being the minimizer of this problem. This new problem can be directly treated numerically by using the Singular Value Decomposition (SVD) method to represent the statically admissible stress fields, along with any unconstrained optimization tool, e.g.: Conjugate Gradient (CG) or Variable Metric (VM) method in multidimensions.

(n + 1)-TENSOR NORMS OF LAPRESTÉ'S TYPE

We study an (n + 1)-tensor norm αr extending to (n + 1)-fold tensor products, the classical one of Lapresté in the case n = 1. We characterise the maps of the minimal and the maximal multi-linear operator ideals related to αr in the sense of Defant and Floret (A. Defant and K. Floret, Tensor norms and operator ideals, North Holland Mathematical Studies, no. 176 (North Holland, Amsterdam, Netherlands, 1993). As an application we give a complete description of the reflexivity of the αr-tensor product (⊗j = 1n + 1 ℓuj, αr).

Local complementation and the extension of bilinear mappings

We study different aspects of the connections between local theory of Banach spaces and the problem of the extension of bilinear forms from subspaces of Banach spaces. Among other results, we prove that if X is not a Hilbert space then one may find a subspace of X for which there is no Aron–Berner extension. We also obtain that the extension of bilinear forms from all the subspaces of a given X forces such X to contain no uniform copies of ℓpn for p ∈ [1, 2). In particular, X must have type 2 − ϵ for every ϵ > 0. Also, we show that the bilinear version of the Lindenstrauss–Pełczyński and Johnson–Zippin theorems fail. We will then consider the notion of locally α-complemented subspace for a reasonable tensor norm α, and study the connections between α-local complementation and the extendability of α*-integral operators.