Related papers: Tensor Ranks for the Pedestrian for Dimension Redu…

Tensors offer a natural representation for many kinds of data frequently encountered in machine learning. Images, for example, are naturally represented as third order tensors, where the modes correspond to height, width, and channels.…

Asymptotic tensor rank is notoriously difficult to determine. Indeed, determining its value for the $2\times 2$ matrix multiplication tensor would determine the matrix multiplication exponent, a long-standing open problem. On the other…

In this paper, we introduce a new tensor decomposition for third order tensors, which decomposes a third order tensor to three third order low rank tensors in a balanced way. We call such a decomposition the triple decomposition, and the…

Low-rank tensor approximation error bounds are proposed for the case of noisy input data that depend on low-rank representation type, rank and the dimensionality of the tensor. The bounds show that high-dimensional low-rank structured…

We develop a framework to analyse invariant decompositions of elements of tensor product spaces. Namely, we define an invariant decomposition with indices arranged on a simplicial complex, and which is explicitly invariant under a group…

We consider the problem of low-rank decomposition of incomplete multiway tensors. Since many real-world data lie on an intrinsically low dimensional subspace, tensor low-rank decomposition with missing entries has applications in many data…

A symmetric tensor of small rank decomposes into a configuration of only few vectors. We study the variety of tensors for which this configuration is a unit norm tight frame.

The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an l-tensor. The tensor product of s and t is a (k + l)-tensor. Tensor rank is…

Higher-order tensors arise frequently in applications such as neuroimaging, recommendation system, social network analysis, and psychological studies. We consider the problem of low-rank tensor estimation from possibly incomplete,…

In tensor completion tasks, the traditional low-rank tensor decomposition models suffer from the laborious model selection problem due to their high model sensitivity. In particular, for tensor ring (TR) decomposition, the number of model…

Tensor decomposition is an important technique for capturing the high-order interactions among multiway data. Multi-linear tensor composition methods, such as the Tucker decomposition and the CANDECOMP/PARAFAC (CP), assume that the complex…

Projective Norms are a class of tensor norms that map on the input and output spaces. These norms are useful for providing a measure of entanglement. Calculating the projective norms is an NP-hard problem, which creates challenges in…

Decompositions of higher-order tensors into sums of simple terms are ubiquitous. We show that in order to verify that two tensors are generated by the same (possibly scaled) terms it is not necessary to compute the individual…

Most currently used tensor regression models for high-dimensional data are based on Tucker decomposition, which has good properties but loses its efficiency in compressing tensors very quickly as the order of tensors increases, say greater…

Continuous-time Markov chains describing interacting processes exhibit a state space that grows exponentially in the number of processes. This state-space explosion renders the computation or storage of the time-marginal distribution, which…

Low-rank approximation is a technique to approximate a tensor or a matrix with a reduced rank to reduce the memory required and computational cost for simulation. Its broad applications include dimension reduction, signal processing,…

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…

The low-rank approximation is a complexity reduction technique to approximate a tensor or a matrix with a reduced rank, which has been applied to the simulation of high dimensional problems to reduce the memory required and computational…

The goal of tensor completion is to fill in missing entries of a partially known tensor under a low-rank constraint. In this paper, we mainly study low rank third-order tensor completion problems by using Riemannian optimization methods on…

Higher-order data with high dimensionality arise in a diverse set of application areas such as computer vision, video analytics and medical imaging. Tensors provide a natural tool for representing these types of data. Although there has…