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We present a novel tensor network algorithm to solve the time-dependent, gray thermal radiation transport equation. The method invokes a tensor train (TT) decomposition for the specific intensity. The efficiency of this approach is dictated…

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,…

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost…

This paper proposes a novel method for learning highly nonlinear, multivariate functions from examples. Our method takes advantage of the property that continuous functions can be approximated by polynomials, which in turn are representable…

Tensor networks are a class of algorithms aimed at reducing the computational complexity of high-dimensional problems. They are used in an increasing number of applications, from quantum simulations to machine learning. Exploiting data…

We consider the solution of linear systems with tensor product structure using a GMRES algorithm. In order to cope with the computational complexity in large dimension both in terms of floating point operations and memory requirement, our…

We present a framework using the Quantized Tensor Train (QTT) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the QTT decomposition can be used as a hierarchical…

Tensor Networks (TN) offer a powerful framework to efficiently represent very high-dimensional objects. TN have recently shown their potential for machine learning applications and offer a unifying view of common tensor decomposition models…

This paper studies a tensor-structured linear regression model with a scalar response variable and tensor-structured predictors, such that the regression parameters form a tensor of order $d$ (i.e., a $d$-fold multiway array) in…

Tensor decomposition is a mathematically supported technique for data compression. It consists of applying some kind of a Low Rank Decomposition technique on the tensors or matrices in order to reduce the redundancy of the data. However, it…

In this article, we derive a semi-Lagrangian scheme for the solution of the Vlasov equation represented as a low-parametric tensor. Grid-based methods for the Vlasov equation have been shown to give accurate results but their use has mostly…

Invariance has recently proven to be a powerful inductive bias in machine learning models. One such class of predictive or generative models are tensor networks. We introduce a new numerical algorithm to construct a basis of tensors that…

Various tensor decomposition methods have been proposed for data compression. In real world applications of the tensor decomposition, selecting the tensor shape for the given data poses a challenge and the shape of the tensor may affect the…

Tensors are often compressed by expressing them in low rank tensor formats. In this paper, we develop three methodologies that bound the compressibility of a tensor: (1) Algebraic structure, (2) Smoothness, and (3) Displacement structure.…

Compressed sensing extends from the recovery of sparse vectors from undersampled measurements via efficient algorithms to the recovery of matrices of low rank from incomplete information. Here we consider a further extension to the…

A tensor is a multi-way array that can represent, in addition to a data set, the expression of a joint law or a multivariate function. As such it contains the description of the interactions between the variables corresponding to each of…

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…

The goals of this work are two-fold: firstly, to propose a new theoretical framework for representing random fields on a large class of multidimensional geometrical domain in the tensor train format; secondly, to develop a new algorithm…

A study on the application of the Tensor Train decomposition method to 3D direct numerical simulation data of channel turbulence flow is presented. The approach is validated with respect to compression rate and storage requirement. In tests…

Tensor methods have become a promising tool to solve high-dimensional problems in the big data era. By exploiting possible low-rank tensor factorization, many high-dimensional model-based or data-driven problems can be solved to facilitate…