Related papers: Tensor Eigenvalue Complementarity Problems
We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be…
Recently, three numerical methods for the computation of eigenvalues of singular matrix pencils, based on a rank-completing perturbation, a rank-projection, or an augmentation were developed. We show that all three approaches can be…
In this article we introduce column adequate tensor in the context of tensor complementarity problem and consider some important properties. The tensor complementarity problem is a class of nonlinear complematarity problems with the…
We study a statistical model for the tensor principal component analysis problem introduced by Montanari and Richard: Given a order-$3$ tensor $T$ of the form $T = \tau \cdot v_0^{\otimes 3} + A$, where $\tau \geq 0$ is a signal-to-noise…
Matrix completion, the problem of completing missing entries in a data matrix with low dimensional structure (such as rank), has seen many fruitful approaches and analyses. Tensor completion is the tensor analog, that attempts to impute…
This paper introduces the notion of tubular eigenvalues of third-order tensors with respect to T-products of tensors and analyzes their properties. A focus of the paper is to discuss relations between tubular eigenvalues and two alternative…
Copositivity of tensors plays an important role in vacuum stability of a general scalar potential, polynomial optimization, tensor complementarity problem and tensor generalized eigenvalue complementarity problem. In this paper, we propose…
In this paper, we consider the {\it generalized polynomial complementarity problem} (GPCP), which covers the recently introduced {\it polynomial complementarity problem} (PCP) and the well studied {\it tensor complementarity problem} (TCP)…
We describe an algorithm to compute the extremal eigenvalues and corresponding eigenvectors of a symmetric matrix by solving a sequence of Quadratic Binary Optimization problems. This algorithm is robust across many different classes of…
In this paper, we mainly develop the well-known vector and matrix polynomial extrapolation methods in tensor framework. To this end, some new products between tensors are defined and the concept of positive definitiveness is extended for…
In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. 2-tensors, such a representation can be obtained…
In this article, we develop methods for estimating a low rank tensor from noisy observations on a subset of its entries to achieve both statistical and computational efficiencies. There have been a lot of recent interests in this problem of…
We theoretically and experimentally investigate tensor-based regression and classification. Our focus is regularization with various tensor norms, including the overlapped trace norm, the latent trace norm, and the scaled latent trace norm.…
This paper studies symmetric tensor decompositions. For symmetric tensors, there exist linear relations of recursive patterns among their entries. Such a relation can be represented by a polynomial, which is called a generating polynomial.…
The Ehrhart polynomial and the reciprocity theorems by Ehrhart \& Macdonald are extended to tensor valuations on lattice polytopes. A complete classification is established of tensor valuations of rank up to eight that are equivariant with…
Identifying locally optimal solutions is an important issue given an optimization model. In this paper, we focus on a special class of symmetric tensors termed regular simplex tensors, which is a newly-emerging concept, and investigate its…
Finding the rank of a tensor is a problem that has many applications. Unfortunately it is often very difficult to determine the rank of a given tensor. Inspired by the heuristics of convex relaxation, we consider the nuclear norm instead of…
We lay the geometric foundations for the study of the characteristic polynomial of tensors. For symmetric tensors of order $d \geq 3$ and dimension $2$ and symmetric tensors of order $3$ and dimension $3$, we prove that only finitely many…
Recently, the eigenvalue problems formulated with symmetric positive definite bilinear forms have been well investigated with the aim of explicit bounds for the eigenvalues. In this paper, the existing theorems for bounding eigenvalues are…
We study linear problems defined on tensor products of Hilbert spaces with an additional (anti-) symmetry property. We construct a linear algorithm that uses finitely many continuous linear functionals and show an explicit formula for its…