Related papers: Eigenschemes of Ternary Tensors

We present a comprehensive construction of scalar, vector and tensor harmonics on maximally symmetric three-dimensional spaces. Our formalism relies on the introduction of spin-weighted spherical harmonics and a generalized helicity basis…

In this paper, we introduce a type of tensor neural network. For the first time, we propose its numerical integration scheme and prove the computational complexity to be the polynomial scale of the dimension. Based on the tensor product…

We present a new framework for computing Z-eigenvectors of general tensors based on numerically integrating a dynamical system that can only converge to a Z-eigenvector. Our motivation comes from our recent research on spacey random walks,…

Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a…

We derive transformation formulas for the generalized polarization tensors under rigid motions and scaling in three dimensions, and use them to construct an infinite number of invariants under those transformations. These invariants can be…

Eigenvalue distributions are important dynamical quantities in matrix models, and it is a challenging problem to derive them in tensor models. In this paper, we consider real symmetric order-three tensors with Gaussian distributions as the…

In this article, we formulate necessary and sufficient polynomial equations for the existence of a symmetry plane or an order-two axial symmetry for a totally symmetric tensor of order n $\ge$ 1. These conditions are effective and of degree…

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…

4x4x3 absolutely nonsingular tensors are characterized by their determinant polynomial. Non-quivalence among absolutely nonsingular tensors with respect to a class of linear transformations, which do not chage the tensor rank,is studied. It…

While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention…

In this paper, the concepts of Pareto $H$-eigenvalue and Pareto $Z$-eigenvalue are introduced for studying constrained minimization problem and the necessary and sufficient conditions of such eigenvalues are given. It is proved that a…

We compute the the Balmer spectra of compact objects of tensor triangulated categories whose objects are filtered or graded objects of (or sheaves valued in) another tensor triangulated category. Notable examples include the filtered…

We consider two hypothesis testing problems for low-rank and high-dimensional tensor signals, namely the tensor signal alignment and tensor signal matching problems. These problems are challenging due to the high dimension of tensors and…

We classify tensors with maximal and next to maximal dimensional symmetry groups under a natural genericity assumption (1-genericity), in dimensions greater than 7. In other words, we classify minimal dimensional orbits in the space of…

We study the symmetric outer product decomposition which decomposes a fully (partially) symmetric tensor into a sum of rank-one fully (partially) symmetric tensors. We present iterative algorithms for the third-order partially symmetric…

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…

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…

In this paper, we focus on developing randomized algorithms for the computation of low multilinear rank approximations of tensors based on the random projection and the singular value decomposition. Following the theory of the singular…

We consider Hecke symmetries on a 3-dimensional vector space with the associated R-symmetric algebra isomorphic to the polynomial algebra $k[x_1,x_2,x_3]$ twisted by an automorphism. The main result states that any such a Hecke symmetry is…

This work investigates the multiplicity and differentiability of eigenfrequencies in structures with various symmetries. In particular, the study explores how the geometric and design variable symmetries affect the distribution of…