scholarly journals Incoherent Tensor Norms and Their Applications in Higher Order Tensor Completion

2017 ◽  
Vol 63 (10) ◽  
pp. 6753-6766 ◽  
Author(s):  
Ming Yuan ◽  
Cun-Hui Zhang
Keyword(s):  
Higher Order ◽  
Tensor Completion ◽  
Order Tensor ◽  
Tensor Norms

scholarly journals Image Completion in Embedded Space Using Multistage Tensor Ring Decomposition

2021 ◽  
Vol 4 ◽  
Author(s):  
Farnaz Sedighin ◽  
Andrzej Cichocki
Keyword(s):  
Data Processing ◽  
Incomplete Data ◽  
State Of The Art ◽  
Sampling Rate ◽  
Higher Order ◽  
Image Completion ◽  
Tensor Completion ◽  
Order Tensor ◽  
Efficient Data ◽  
Very High

Tensor Completion is an important problem in big data processing. Usually, data acquired from different aspects of a multimodal phenomenon or different sensors are incomplete due to different reasons such as noise, low sampling rate or human mistake. In this situation, recovering the missing or uncertain elements of the incomplete dataset is an important step for efficient data processing. In this paper, a new completion approach using Tensor Ring (TR) decomposition in the embedded space has been proposed. In the proposed approach, the incomplete data tensor is first transformed into a higher order tensor using the block Hankelization method. Then the higher order tensor is completed using TR decomposition with rank incremental and multistage strategy. Simulation results show the effectiveness of the proposed approach compared to the state of the art completion algorithms, especially for very high missing ratios and noisy cases.

scholarly journals Scaled Coupled Norms and Coupled Higher-Order Tensor Completion

2020 ◽  
Vol 32 (2) ◽  
pp. 447-484
Author(s):  
Kishan Wimalawarne ◽  
Makoto Yamada ◽  
Hiroshi Mamitsuka
Keyword(s):  
Real Data ◽  
Higher Order ◽  
Low Rank ◽  
Tensor Completion ◽  
Coupled Modes ◽  
Coupled Mode ◽  
The Matrix ◽  
Tensor Norms ◽  
Higher Order Tensors ◽  
Risk Bounds

Recently, a set of tensor norms known as coupled norms has been proposed as a convex solution to coupled tensor completion. Coupled norms have been designed by combining low-rank inducing tensor norms with the matrix trace norm. Though coupled norms have shown good performances, they have two major limitations: they do not have a method to control the regularization of coupled modes and uncoupled modes, and they are not optimal for couplings among higher-order tensors. In this letter, we propose a method that scales the regularization of coupled components against uncoupled components to properly induce the low-rankness on the coupled mode. We also propose coupled norms for higher-order tensors by combining the square norm to coupled norms. Using the excess risk-bound analysis, we demonstrate that our proposed methods lead to lower risk bounds compared to existing coupled norms. We demonstrate the robustness of our methods through simulation and real-data experiments.

A Novel Regularized Model for Third-Order Tensor Completion

2021 ◽  
pp. 1-1
Author(s):  
Yi Yang ◽  
Lixin Han ◽  
Yuanzhen Liu ◽  
Jun Zhu ◽  
Hong Yan
Keyword(s):  
Tensor Completion ◽  
Third Order ◽  
Order Tensor ◽  
Regularized Model

Assessing the Use of Tensor Closure Methods With Orientation Distribution Reconstruction Functions

Materials ◽  
2003 ◽  
Author(s):  
David A. Jack ◽  
Douglas E. Smith
Keyword(s):  
Distribution Function ◽  
Fourth Order ◽  
Lower Order ◽  
Series Representation ◽  
Orientation Distribution ◽  
Higher Order ◽  
Interaction Coefficients ◽  
Order Tensor ◽  
Orientation Tensor ◽  
Orientation Tensors

Orientation tensors are widely used to describe fiber distri-butions in short fiber reinforced composite systems. Although these tensors capture the stochastic nature of concentrated fiber suspensions in a compact form, the evolution equation for each lower order tensor is a function of the next higher order tensor. Flow calculations typically employ a closure that approximates the fourth-order orientation tensor as a function of the second order orientation tensor. Recent work has been done with eigen-value based and invariant based closure approximations of the fourth-order tensor. The effect of using lower order tensors tensors in process simulations by reconstructing the distribution function from successively higher order orientation tensors in a Fourier series representation is considered. This analysis uses the property that orientation tensors are related to the series expansion coefficients of the distribution function. Errors for several closures are investigated and compared with errors developed when using a reconstruction from the exact 2nd, 4th, and 6th order orientation tensors over a range of interaction coefficients from 10−4 to 10−1 for several flow fields.

scholarly journals Inversion of higher order isotropic tensors with minor symmetries and solution of higher order heterogeneity problems

2010 ◽  
Vol 467 (2126) ◽  
pp. 314-332 ◽  
Author(s):  
Vincent Monchiet ◽  
Guy Bonnet
Keyword(s):  
Closed Form ◽  
Strain Field ◽  
Spherical Inclusion ◽  
Higher Order ◽  
Quadratic Polynomial ◽  
Elastic Matrix ◽  
Closed Form Expression ◽  
Form Expression ◽  
Order Tensor ◽  
Strain Fields

In this paper, the derivation of irreducible bases for a class of isotropic 2 n th-order tensors having particular ‘minor symmetries’ is presented. The methodology used for obtaining these bases consists of extending the concept of deviatoric and spherical parts, commonly used for second-order tensors, to the case of an n th-order tensor. It is shown that these bases are useful for effecting the classical tensorial operations and especially the inversion of a 2 n th-order tensor. Finally, the formalism introduced in this study is applied for obtaining the closed-form expression of the strain field within a spherical inclusion embedded in an infinite elastic matrix and subjected to linear or quadratic polynomial remote strain fields.

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