Related papers: On low-rank approximability of solutions to high-d…
The low rank tensor approximation problem (LRTAP) is to find a tensor whose rank is small and that is close to a given one. This paper studies the LRTAP when the tensor to be approximated is close to a low rank one. Both symmetric and…
A lower semi-definite self-adjoint linear operator in a Hilbert space is taken whose discrete spectrum is not empty and comprises at least several eigenvalues $\lambda_{min}=\lambda_1\leqslant\ldots\leqslant\lambda_m<\sigma_{ess}$. The…
Low-rank approximation of tensors has been widely used in high-dimensional data analysis. It usually involves singular value decomposition (SVD) of large-scale matrices with high computational complexity. Sketching is an effective data…
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…
There has been continued interest in seeking a theorem describing optimal low-rank approximations to tensors of order 3 or higher, that parallels the Eckart-Young theorem for matrices. In this paper, we argue that the naive approach to this…
This paper is concerned with the approximation of tensors using tree-based tensor formats, which are tensor networks whose graphs are dimension partition trees. We consider Hilbert tensor spaces of multivariate functions defined on a…
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…
We propose inexact subspace iteration for solving high-dimensional eigenvalue problems with low-rank structure. Inexactness stems from low-rank compression, enabling efficient representation of high-dimensional vectors in a low-rank tensor…
For a given symmetric tensor, we aim at finding a new one whose symmetric rank is small and that is close to the given one. There exist linear relations among the entries of low rank symmetric tensors. Such linear relations can be expressed…
In this paper, we introduce a method for multivariate function approximation using function evaluations, Chebyshev polynomials, and tensor-based compression techniques via the Tucker format. We develop novel randomized techniques to…
We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to infinite-dimensional feature spaces,…
In this paper we propose efficient randomized fixed-precision techniques for low tubal rank approximation of tensors. The proposed methods are faster and more efficient than the existing fixed-precision algorithms for approximating the…
Modeling of multidimensional signal using tensor is more convincing than representing it as a collection of matrices. The tensor based approaches can explore the abundant spatial and temporal structures of the mutlidimensional signal. The…
We propose a low-rank tensor approach to approximate linear transport and nonlinear Vlasov solutions and their associated flow maps. The approach takes advantage of the fact that the differential operators in the Vlasov equation is tensor…
There exist linear relations among tensor entries of low rank tensors. These linear relations can be expressed by multi-linear polynomials, which are called generating polynomials. We use generating polynomials to compute tensor rank…
We present an efficient low-rank approximation algorithm for non-negative tensors. The algorithm is derived from our two findings: First, we show that rank-1 approximation for tensors can be viewed as a mean-field approximation by treating…
We propose an isogeometric solver for Poisson problems that combines i)low-rank tensor techniques to approximate the unknown solution and the system matrix, as a sum of a few terms having Kronecker product structure, ii) a Truncated…
Low-rank modeling has many important applications in computer vision and machine learning. While the matrix rank is often approximated by the convex nuclear norm, the use of nonconvex low-rank regularizers has demonstrated better empirical…
We consider the problem of approximating an affinely structured matrix, for example a Hankel matrix, by a low-rank matrix with the same structure. This problem occurs in system identification, signal processing and computer algebra, among…
The numerical solution of partial differential equations on high-dimensional domains gives rise to computationally challenging linear systems. When using standard discretization techniques, the size of the linear system grows exponentially…