Semialgebraic Geometry of Nonnegative Tensor Rank

The compression method, first proposed in 2012, is based on the non-negative tensor decomposition for interference hyper-spectral image data. As a tensor is generated by a huge amount of interference hyper-spectral images, the multiplicative update algorithm is made extremely complicated, and even unfeasible. To reduce the computational cost and speed up the convergence, this paper, based on the characteristics of interference hyper-spectral images, develops a new algorithm using different down-sampling factors for different non-negative wavelet sub-band tensors. The experimental results showed that this algorithm could significantly shorten the running time, while maintaining a good compression performance compared with the conventional methods.

On the uniqueness of the positive Z-eigenvector for nonnegative tensors

Eigenvalue distributions are important dynamic quantities in matrix models, and it is an interesting challenge to study corresponding quantities in tensor models. We study real tensor eigenvalue/vector distributions for real symmetric order-three random tensors with a Gaussian distribution as the simplest case. We first rewrite this problem as the computation of a partition function of a four-fermi theory with R replicated fermions. The partition function is exactly computed for some small-N,R cases, and is shown to precisely agree with Monte Carlo simulations. For large-N, it seems difficult to compute it exactly, and we apply an approximation using a self-consistency equation for two-point functions and obtain an analytic expression. It turns out that the real tensor eigenvalue distribution obtained by taking R = 1/2 is simply the Gaussian within this approximation. We compare the approximate expression with Monte Carlo simulations, and find that, if an extra overall factor depending on N is multiplied to the the expression, it agrees well with the Monte Carlo results. It is left for future study to improve the approximation for large-N to correctly derive the overall factor.

We study the real and complex geometric simplicity of nonnegative irreducible tensors. First, we prove some basic conclusions. Based on the conclusions, the real geometric simplicity of the spectral radius of an evenorder nonnegative irreducible tensor is proved. For an odd-order nonnegative irreducible tensor, sufficient conditions are investigated to ensure the spectral radius to be real geometrically simple. Furthermore, the complex geometric simplicity of nonnegative irreducible tensors is also studied.

The main purpose of this paper is to consider the Perron pair of an irreducible and symmetric nonnegative tensor and the smallest eigenvalue of an irreducible and symmetric nonsingular \(\mathscr {M}\)-tensor. We analyze the analytical property of an algebraic simple eigenvalue of symmetric tensors. We also derive an inequality about the Perron pair of nonnegative tensors based on plane stochastic tensors. We finally consider the perturbation of the smallest eigenvalue of nonsingular \(\mathscr {M}\)-tensors and design a strategy to compute its smallest eigenvalue. We verify our results via random numerical examples.

In this paper, we present an algorithm that allows us to compute the permanent of a tensor by using Laplace expansion. We prove that the permanent of a $4$-dimensional polystochastic $(0,1)$-tensor of order $n$ constructed using a special $n\times (n-1)$ row-Latin rectangle $R$ with no transversals is positive. Also, we show that the permanent of an even-dimensional polystochastic $(0,1)$-tensor of order $n$ constructed using the row-Latin rectangle $R$ is positive. The result obtained here proves that each odd-dimensional Latin hypercube of order $4$ has a transversal (Wanless' conjecture for odd-dimensional Latin hypercubes of order $4$). We prove that the number of perfect matchings of the bipartite hypergraph associated to an even-dimensional polystochastic $(0,1)$-tensor of order $4$ is positive. Furthermore, we extend some results concerning polystochastic $(0,1)$-tensors to nonnegative polystochastic tensors. Moreover, we prove that the permanent of a $ 4 $-dimensional nonnegative polystochastic tensor of order $n$ constructed using the row-Latin rectangle $R$ is positive. More generally, we show that the permanent of an even-dimensional nonnegative polystochastic tensor of order $n$ constructed using the row-Latin rectangle $R$ is positive. The result obtained here proves that the permanent of an even-dimensional nonnegative polystochastic tensor of order $4$ is positive.

The main aim of this paper is to develop a new algorithm for computing a nonnegative low multi‐rank tensor approximation for a nonnegative tensor. In the literature, there are several nonnegative tensor factorizations or decompositions, and their approaches are to enforce the nonnegativity constraints in the factors of tensor factorizations or decompositions. In this paper, we study nonnegativity constraints in tensor entries directly, and a low rank approximation for the transformed tensor by using discrete Fourier transformation matrix, discrete cosine transformation matrix, or unitary transformation matrix. This strategy is particularly useful in imaging science as nonnegative pixels appear in tensor entries and a low rank structure can be obtained in the transformation domain. We propose an alternating projections algorithm for computing such a nonnegative low multi‐rank tensor approximation. The convergence of the proposed projection method is established. Numerical examples for multidimensional images are presented to demonstrate that the performance of the proposed method is better than that of nonnegative low Tucker rank tensor approximation and the other nonnegative tensor factorizations and decompositions.

It is shown that the relative distance in Frobenius norm of a real symmetric order-d tensor of rank-two to its best rank-one approximation is upper bounded by sqrt{1-(1-1/d)^{d-1}}. This is achieved by determining the minimal possible ratio between spectral and Frobenius norm for symmetric tensors of border rank two, which equals left( 1-{1}/{d}right) ^{(d-1)/{2}}. These bounds are also verified for arbitrary real rank-two tensors by reducing to the symmetric case.

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 simplest case, and derive an explicit formula for signed distributions of real tensor eigenvectors: Each real tensor eigenvector contributes to the distribution by ±1, depending on the sign of the determinant of an associated Hessian matrix. The formula is expressed by the confluent hypergeometric function of the second kind, which is obtained by computing a partition function of a four-fermi theory. The formula can also serve as lower bounds of real eigenvector distributions (with no signs), and their tightness/looseness are discussed by comparing with Monte Carlo simulations. Large-N limits are taken with the characteristic oscillatory behavior of the formula being preserved.

A real tensor T of rank r is identifiable if it has a unique decomposition with rank 1 tensors. Sometimes identifiability fails over , for general tensors of fixed rank. This behaviour is peculiar when r is sub-generic. In several cases, the failure is due to an elliptic normal curve passing through general points of the corresponding variety of rank 1 tensors (Segre, Veronese, Grassmann). When this happens, we prove the existence of nonempty euclidean open subsets of some varieties of rank r tensors, whose elements have several complex decompositions, but only one of them is real.

Mobility mining has lots of applications in urban planning and transportation systems. In particular, extracting mobility patterns enables service providers to have a global insight about the mobility behaviors which consequently leads to providing better services to the citizens. In the recent years several data mining techniques have been presented to tackle this problem. These methods usually are either spatial extension of temporal methods or temporal extension of spatial methods. However, still a framework that can keep the natural structure of mobility data has not been considered. Non-negative tensor factorizations (NNTF) have shown great applications in topic modelling and pattern recognition. However, unfortunately their usefulness in mobility mining is less explored. In this paper we propose a new mobility pattern mining framework based on a recent non-negative tensor model called BetaNTF. We also present a new approach based on interpretability concept for determination of number of components in the tensor rank selection process. We later demonstrate some meaningful mobility patterns extracted with the proposed method from bike sharing network mobility data in Boston, USA.

In this article, the index of imprimitivity of an irreducible nonnegative matrix in the famous Perron-Frobenius theorem is studied within a more general framework, both in a more general tensor setting and in a more natural spectral symmetry perspective. A $k$-th order tensor has symmetric spectrum if the set of eigenvalues is symmetric under a group action with the group being a subgroup of the multiplicative group of $k$-th roots of unity.A sufficient condition, in terms of linear equations over the quotient ring, for a tensor possessing symmetric spectrum is given, which becomes also necessary when the tensor is nonnegative, symmetric and weakly irreducible, or an irreducible nonnegative matrix. Moreover, it is shown that for a weakly irreducible nonnegative tensor, the spectral symmetries are the same when either counting or ignoring multiplicities of the eigenvalues.In particular, the spectral symmetry (index of imprimitivity) of an irreducible nonnegative Sylvester matrix is completely resolved via characterizations with the indices of its positive entries. It is shown that the spectrum of an irreducible nonnegative Sylvester matrix can only be $1$-symmetric or $2$-symmetric, and the exact situations are fully described. With this at hand, the spectral symmetry of a nonnegative two-dimensional symmetric tensor with arbitrary order is also completely characterized.

The conventional tensor-network states employ real-space product states as reference wave functions. Here, we propose a many-variable variational Monte Carlo (mVMC) method combined with tensor networks by taking advantages of both to study fermionic models. The variational wave function is composed of a pair product wave function operated by real space correlation factors and tensor networks. Moreover, we can apply quantum number projections, such as spin, momentum and lattice symmetry projections, to recover the symmetry of the wave function to further improve the accuracy. We benchmark our method for one- and two-dimensional Hubbard models, which show significant improvement over the results obtained individually either by mVMC or by tensor network. We have applied the present method to hole doped Hubbard model on the square lattice, which indicates the stripe charge/spin order coexisting with a weak $d$-wave superconducting order in the ground state for the doping concentration less than 0.3, where the stripe oscillation period gets longer with increasing hole concentration. The charge homogeneous and highly superconducting state also exists as a metastable excited state for the doping concentration less than 0.25.

Tensor Rank One Discriminant Analysis—A convergent method for discriminative multilinear subspace selection

Nonconvex low-rank tensor approximation with graph and consistent regularizations for multi-view subspace learning