The topological structure of scalar, vector, and second-order tensor fields provides an important mathematical basis for data analysis and visualization. In this paper, we extend this framework towards higher-order tensors. First, we establish formal uniqueness properties for a geometrically constrained tensor decomposition. This allows us to define and visualize topological structures in symme...

In this paper, we aim at the problem of tensor data completion. Tensor-train decomposition is adopted because of its powerful representation ability and linear scalability to tensor order. We propose an algorithm named Sparse Tensortrain Optimization (STTO) which considers incomplete data as sparse tensor and uses first-order optimization method to find the factors of tensor-train decomposition...

This paper presents an algorithm to solve linear systems expressed by a matrix stored in a tensor product format. The proposed solution is based on a LU decomposition of the matrix keeping the tensor product structure. It is shown that the complexity of the decomposition is negligible and the backward and forward substitutions are no more complex than two standard vector-matrices multiplication...

In a variety of situations, integrals of products of eigenfunctions have faster decay than smoothness entails. This phenomenon does not appear for abelian or compact groups, since irreducibles are finite-dimensional, so the decomposition of a tensor product of irreducibles is finite. In contrast, for non-compact, nonabelian groups irreducibles are typically infinite-dimensional, and the decompo...

In this paper we provide a general condition for the reducibility of the ReshetikhinTuraev quantum representations of the mapping class groups. Namely, for any modular tensor category with a special symmetric Frobenius algebra with a non-trivial genus one partition function, we prove that the quantum representations of all the mapping class groups built from the modular tensor category are redu...

A numerical method is proposed to solve the full-Eulerian time-dependent Vlasov-Poisson system in high dimension. The algorithm relies on the construction of a tensor decomposition of the solution whose rank is adapted at each time step. This decomposition is obtained through the use of an efficient modified Progressive Generalized Decomposition (PGD) method, whose convergence is proved. We sug...

Abstract We give a direct tensor decomposition for any density matrix into Hermitian operators. Based upon the decomposition we study when the mixed states are separable and generalize the separability indicators to multi-partite states and show that a density operator is separable if and only if the separable indicator is non-negative. We then derive two bounds for the separable indicator in t...

We discuss extended definitions of linear and multilinear operations such as Kronecker, Hadamard, and contracted products, and establish links between them for tensor calculus. Then we introduce effective low-rank tensor approximation techniques including Candecomp/Parafac (CP), Tucker, and tensor train (TT) decompositions with a number of mathematical and graphical representations. We also pro...

We propose a constructive algorithm, called the tensor-based Kronecker product (KP) singular value decomposition (TKPSVD), that decomposes an arbitrary real matrix A into a finite sum of KP terms with an arbitrary number of d factors, namely A = ∑R j=1 σj A dj ⊗ · · · ⊗A1j . The algorithm relies on reshaping and permuting the original matrix into a d-way tensor, after which its tensor-train ran...

Tensor CANDECOMP/PARAFAC (CP) decomposition is a powerful but computationally challenging tool in modern data analytics. In this paper, we show ways of sampling intermediate steps of alternating minimization algorithms for computing low rank tensor CP decompositions, leading to the sparse alternating least squares (SPALS) method. Specifically, we sample the Khatri-Rao product, which arises as a...