相关论文: Tensor models from the viewpoint of matrix models:…

In this paper, we consider the problem of distributed parameter estimation in sensor networks. Each sensor makes successive observations of an unknown $d$-dimensional parameter, which might be subject to Gaussian random noises. The sensors…

We compute the expected value of powers of the geometric condition number of random tensor rank decompositions. It is shown in particular that the expected value of the condition number of $n_1\times n_2 \times 2$ tensors with a random…

Sparse incidence tensors can represent a variety of structured data. For example, we may represent attributed graphs using their node-node, node-edge, or edge-edge incidence matrices. In higher dimensions, incidence tensors can represent…

This short note presents upper bounds of the expectations of the largest singular values/eigenvalues of various types of random tensors in the non-asymptotic sense. For a standard Gaussian tensor of size $n_1\times\cdots\times n_d$, it is…

Abstract. The purpose of this paper is twofold. We introduce the theory of random tensors, which naturally extends the method of random averaging operators in our earlier work arXiv:1910.08492, to study the propagation of randomness under…

We propose a new definition of characteristic polynomials of tensors based on a partition function of Grassmann variables. This new notion of characteristic polynomial addresses general tensors including totally antisymmetric ones, but not…

We present a new method for online prediction and learning of tensors ($N$-way arrays, $N >2$) from sequential measurements. We focus on the specific case of 3-D tensors and exploit a recently developed framework of structured tensor…

Tensor-valued and matrix-valued measurements of different physical properties are increasingly available in material sciences and medical imaging applications. The eigenvalues and eigenvectors of such multivariate data provide novel and…

Tensors are widely used to represent multiway arrays of data. The recovery of missing entries in a tensor has been extensively studied, generally under the assumption that entries are missing completely at random (MCAR). However, in most…

We study eigenvectors in the deformed Gaussian unitary ensemble of random matrices $H=W\tilde{H}W$, where $\tilde{H}$ is a random matrix from Gaussian unitary ensemble and $W$ is a deterministic diagonal matrix with positive entries. Using…

In this paper, we develop the notion of entropy for uniform hypergraphs via tensor theory. We employ the probability distribution of the generalized singular values, calculated from the higher-order singular value decomposition of the…

Orthogonal decomposition of tensors is a generalization of the singular value decomposition of matrices. In this paper, we study the spectral theory of orthogonally decomposable tensors. For such a tensor, we give a description of its…

Representation theory provides a suitable framework to count and classify invariants in tensor models. We show that there are two natural ways of counting invariants, one for arbitrary rank of the gauge group and a second, which is only…

A real number $\lambda$ is called a Z-eigenvalue of a tensor $A$, if $\lambda$ is an eigenvalue of $A$ and the corresponding eigenvector $v$ is real and satisfies $v^Tv=1$. In this paper we compute the expected number of Z-eigenvalues of a…

This article first introduces the notion of weighted singular value decomposition (WSVD) of a tensor via the Einstein product. The WSVD is then used to compute the weighted Moore-Penrose inverse of an arbitrary-order tensor. We then define…

Dimensionality reduction for high-order tensors is a challenging problem. In conventional approaches, higher order tensors are `vectorized` via Tucker decomposition to obtain lower order tensors. This will destroy the inherent high-order…

Tensor networks are a powerful modeling framework developed for computational many-body physics, which have only recently been applied within machine learning. In this work we utilize a uniform matrix product state (u-MPS) model for…

We study the regularity of densities of distributions that are polynomial images of the standard Gaussian measure on $\mathbb{R}^n$. We assume that the degree of a polynomial is fixed and that each variable enters to a power bounded by…

This paper takes a different approach for the distributed linear parameter estimation over a multi-agent network. The parameter vector is considered to be stochastic with a Gaussian distribution. The sensor measurements at each agent are…

We study distributions of random vectors whose components are second order polynomials in Gaussian random variables. Assuming that the law of such a vector is not absolutely continuous with respect to Lebesgue measure, we derive some…