Related papers: Low-rank tensor approximation for Chebyshev interp…

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…

Low-rank tensor methods for the approximate solution of second-order elliptic partial differential equations in high dimensions have recently attracted significant attention. A critical issue is to rigorously bound the error of such…

The tensor train (TT) rank has received increasing attention in tensor completion due to its ability to capture the global correlation of high-order tensors ($\textrm{order} >3$). For third order visual data, direct TT rank minimization has…

Accurately evaluating configurational integrals for dense solids remains a central and difficult challenge in the statistical mechanics of condensed systems. Here, we present a novel tensor network approach that reformulates the…

In this paper, we introduce a new interpolation scheme to approximate the density of states (DOS) for a class of rank-structured matrices with application to the Tamm-Dancoff approximation (TDA) of the Bethe-Salpeter equation (BSE). The…

We describe a simple, black-box compression format for tensors with a multiscale structure. By representing the tensor as a sum of compressed tensors defined on increasingly coarse grids, we capture low-rank structures on each grid-scale,…

It is often possible to perform reduced order modelling by specifying linear subspace which accurately captures the dynamics of the system. This approach becomes especially appealing when linear subspace explicitly depends on parameters of…

We extend the univariate Newton interpolation algorithm to arbitrary spatial dimensions and for any choice of downward-closed polynomial space, while preserving its quadratic runtime and linear storage cost. The generalisation supports any…

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…

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…

We consider rank-one non-symmetric tensor estimation and derive simple formulas for the mutual information. We start by the order 2 problem, namely matrix factorization. We treat it completely in a simpler fashion than previous proofs using…

In this paper, we introduce a tensor neural network based machine learning method for solving the elliptic partial differential equations with random coefficients in a bounded physical domain. With the help of tensor product structure, we…

Such problems as computation of spectra of spin chains and vibrational spectra of molecules can be written as high-dimensional eigenvalue problems, i.e., when the eigenvector can be naturally represented as a multidimensional tensor. Tensor…

We apply methods and techniques of tropical optimization to develop a new theoretical and computational framework for the implementation of the Analytic Hierarchy Process in multi-criteria problems of rating alternatives from pairwise…

Unlike the matrix case, computing low-rank approximations of tensors is NP-hard and numerically ill-posed in general. Even the best rank-1 approximation of a tensor is NP-hard. In this paper, we use convex optimization to develop…

In this work, we perform Bayesian inference tasks for the chemical master equation in the tensor-train format. The tensor-train approximation has been proven to be very efficient in representing high dimensional data arising from the…

Inspired by a series of remarkable papers in recent years that use Deep Neural Nets to substantially speed up the calibration of pricing models, we investigate the use of Chebyshev Tensors instead of Deep Neural Nets. Given that Chebyshev…

This paper studies the low-rank property of the inverse of a class of large-scale structured matrices in the tensor-train (TT) format, which is typically discretized from differential operators. An interesting question that we are concerned…

The numerical integration of stiff equations is a challenging problem that needs to be approached by specialized numerical methods. Exponential integrators form a popular class of such methods since they are provably robust to stiffness and…

Tensor train decomposition is a powerful tool for dealing with high-dimensional, large-scale tensor data, which is not suffering from the curse of dimensionality. To accelerate the calculation of the auxiliary unfolding matrix, some…