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Higher-order tensors appear in various areas of mechanics as well as physics, medicine or earth sciences. As these tensors are highly complex, most are not well understood. Thus, the analysis and the visualization process form a highly…

Tensors of order three or higher have found applications in diverse fields, including image and signal processing, data mining, biomedical engineering and link analysis, to name a few. In many applications that involve for example time…

The tensor train decomposition decomposes a tensor into a "train" of 3-way tensors that are interconnected through the summation of auxiliary indices. The decomposition is stable, has a well-defined notion of rank and enables the user to…

One of the main issues in computing a tensor decomposition is how to choose the number of rank-one components, since there is no finite algorithms for determining the rank of a tensor. A commonly used approach for this purpose is to find a…

Decompositions of tensors into factor matrices, which interact through a core tensor, have found numerous applications in signal processing and machine learning. A more general tensor model which represents data as an ordered network of…

In this paper, a new definition of tensor p-shrinkage nuclear norm (p-TNN) is proposed based on tensor singular value decomposition (t-SVD). In particular, it can be proved that p-TNN is a better approximation of the tensor average rank…

The existing tensor networks adopt conventional matrix product for connection. The classical matrix product requires strict dimensionality consistency between factors, which can result in redundancy in data representation. In this paper,…

This study aims to solve the over-reliance on the rank estimation strategy in the standard tensor factorization-based tensor recovery and the problem of a large computational cost in the standard t-SVD-based tensor recovery. To this end, we…

By a tensor we mean an element of a tensor product of vector spaces over a field. Up to a choice of bases in factors of tensor products, every tensor may be coordinatized, that is, represented as an array consisting of numbers. This note is…

Tensor completion can estimate missing values of a high-order data from its partially observed entries. Recent works show that low rank tensor ring approximation is one of the most powerful tools to solve tensor completion problem. However,…

Recently, the \textit{Tensor Nuclear Norm~(TNN)} regularization based on t-SVD has been widely used in various low tubal-rank tensor recovery tasks. However, these models usually require smooth change of data along the third dimension to…

Due to the explosive growth of large-scale data sets, tensors have been a vital tool to analyze and process high-dimensional data. Different from the matrix case, tensor decomposition has been defined in various formats, which can be…

Machine learning and data mining algorithms are becoming increasingly important in analyzing large volume, multi-relational and multi--modal datasets, which are often conveniently represented as multiway arrays or tensors. It is therefore…

In this paper we focus on the problem of completion of multidimensional arrays (also referred to as tensors) from limited sampling. Our approach is based on a recently proposed tensor-Singular Value Decomposition (t-SVD) [1]. Using this…

The tensor Singular Value Decomposition (t-SVD) for third order tensors that was proposed by Kilmer and Martin~\cite{2011kilmer} has been applied successfully in many fields, such as computed tomography, facial recognition, and video…

DeepTensor is a computationally efficient framework for low-rank decomposition of matrices and tensors using deep generative networks. We decompose a tensor as the product of low-rank tensor factors (e.g., a matrix as the outer product of…

Tensors with unit Frobenius norm are fundamental objects in many fields, including scientific computing and quantum physics, which are able to represent normalized eigenvectors and pure quantum states. While the tensor train decomposition…

The (efficient and parsimonious) decomposition of higher-order tensors is a fundamental problem with numerous applications in a variety of fields. Several methods have been proposed in the literature to that end, with the Tucker and PARAFAC…

The theory and computation of tensors with different tensor products play increasingly important roles in scientific computing and machine learning. Different products aim to preserve different algebraic properties from the matrix algebra,…

A novel tensor decomposition framework, termed Tensor Star (TS) decomposition, is proposed which represents a new type of tensor network decomposition based on tensor contractions. This is achieved by connecting the core tensors in a ring…