We study the problem of nonnegative rank-one approximation of a nonnegative tensor, and show that the globally optimal solution that minimizes the generalized Kullback-Leibler divergence can be efficiently obtained, i.e., it is not NP-hard. This result works for arbitrary nonnegative tensors with an arbitrary number of modes (including two, i.e., matrices). We derive a closed-form expression fo...

The positive mass theorem in general relativity states that an asymptotically flat spacetime, if the momentum–energy tensor is divergence-free and satisfies a dominant energy condition, then total four-vector can be formed, of which component nonnegative. In this paper, we take wave four-tensor plane light free space as counterexample to show there no guarantee formed. Thus theoretical framewor...

By estimating the ratio of the smallest component and the largest component of a Perron vector, we provide a new bound for the spectral radius of a nonnegative tensor. And it is proved that the proposed result improves the bound in (Li and Ng in Numer. Math. 130(2):315-335, 2015).

Tensor networks are a central tool in condensed matter physics. In this paper, we initiate the study of tensor network non-zero testing (TNZ): Given a tensor network T , does T represent a non-zero vector? We show that TNZ is not in the Polynomial-Time Hierarchy unless the hierarchy collapses. We next show (among other results) that the special cases of TNZ on nonnegative and injective tensor n...

Nonnegative tensor factorization (NTF) is a technique for computing a parts-based representation of high-dimensional data. NTF excels at exposing latent structures in datasets, and at finding good low-rank approximations to the data. We describe an approach for computing the NTF of a dataset that relies only on iterative linear-algebra techniques and that is comparable in cost to the nonnegativ...

In the paper we present new Alternating Least Squares (ALS) algorithms for Nonnegative Matrix Factorization (NMF) and their extensions to 3D Nonnegative Tensor Factorization (NTF) that are robust in the presence of noise and have many potential applications, including multi-way Blind Source Separation (BSS), multi-sensory or multi-dimensional data analysis, and nonnegative neural sparse coding....

A subarea of low-rank approximation that is not covered in this overview is tensors low-rank approximation [WVB10, LV00]. Tensor methods are used in higher order statistical signal processing problems, such as independent component analysis, and multidimensional signal processing, such as spatiotemporal modeling and video processing, to name a few. Other areas of research on low-rank approximat...

Matrix factorisation models have had an explosive growth in popularity in the last decade. It has become popular due to its usefulness in clustering and missing values prediction. We review the main literature for matrix factorisation, focusing on nonnegative matrix factorisation and probabilistic approaches. We also consider several extensions: matrix tri-factorisation, Tensor factorisation, T...

We study M-tensors and various properties of M-tensors are given. Specially, we show that the smallest real eigenvalue of M-tensor is positive corresponding to a nonnegative eigenvector. We propose an algorithm to find the smallest positive eigenvalue and then apply the property to study the positive definiteness of a multivariate form.

We define a generalized mass for asymptotically flat manifolds using some higher order symmetric function of the curvature tensor. This mass is nonnegative when the manifold is locally conformally flat and the σk curvature vanishes at infinity. In addition, with the above assumptions, if the mass is zero, then, near infinity, the manifold is isometric to a Euclidean end.