Related papers: Low Rank Symmetric Tensor Approximations

For the antisymmetric tensors the paper examines a low-rank approximation which is represented via only three vectors. We describe a suitable low-rank format and propose an alternating least squares structure-preserving algorithm for…

Low-rank tensor approximations have shown great potential for uncertainty quantification in high dimensions, for example, to build surrogate models that can be used to speed up large-scale inference problems (Eigel et al., Inverse Problems…

We consider the problem of determining the symmetric tensor rank for symmetric tensors with an algebraic geometry approach. We give algorithms for computing the symmetric rank for $2\times ... \times 2$ tensors and for tensors of small…

In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal…

Rank-metric codes are subspaces of matrices over finite fields endowed with the rank metric and admit a natural tensorial representation. The tensor rank provides a measure of the minimal size of a decomposition of a code into rank-one…

In this paper, we investigate the generalized low rank approximation to the symmetric positive semidefinite matrix in the Frobenius norm: $$\underset{ rank(X)\leq k}{\min} \sum^m_{i=1}\left \Vert A_i - B_i XB_i^T \right \Vert^2_F,$$ where…

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…

A symmetric tensor may be regarded as a partially symmetric tensor in several different ways. These produce different notions of rank for the symmetric tensor which are related by chains of inequalities. By exploiting algebraic tools such…

Tensor network contractions are widely used in statistical physics, quantum computing, and computer science. We introduce a method to efficiently approximate tensor network contractions using low-rank approximations, where each intermediate…

We consider the synthesis problem of Compressed Sensing - given s and an MXn matrix A, extract from it an mXn submatrix A', certified to be s-good, with m as small as possible. Starting from the verifiable sufficient conditions of…

These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will…

Higher-order low-rank tensor arises in many data processing applications and has attracted great interests. Inspired by low-rank approximation theory, researchers have proposed a series of effective tensor completion methods. However, most…

The tensor rank decomposition is a useful tool for the geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able…

We propose a new technique for constructing low-rank approximations of matrices that arise in kernel methods for machine learning. Our approach pairs a novel automatically constructed analytic expansion of the underlying kernel function…

In this paper, we investigate the sample complexity of recovering tensors with low symmetric rank from symmetric rank-one measurements. This setting is particularly motivated by the study of higher-order interactions and the analysis of…

The Rank Minimization Problem asks to find a matrix of lowest rank inside a linear variety of the space of n x n matrices. The Low Rank Matrix Completion problem asks to complete a partially filled matrix such that the resulting matrix has…

In this paper, we show that the low rank matrix completion problem can be reduced to the problem of finding the rank of a certain tensor.

Low rank tensor learning, such as tensor completion and multilinear multitask learning, has received much attention in recent years. In this paper, we propose higher order matching pursuit for low rank tensor learning problems with a convex…

We study low T-phase-rank approximation of sectorial third-order tensors $\mathscr{A}\in\mathbb{C}^{n\times n\times p}$ under the tensor T-product. We introduce canonical T-phases and T-phase rank, and formulate the approximation task as…

The goal of this work is to fill a gap in [Yang, SIAM J. Matrix Anal. Appl, 41 (2020), 1797--1825]. In that work, an approximation procedure was proposed for orthogonal low-rank tensor approximation; however, the approximation lower bound…