Conjugate partial-symmetric (CPS) tensors are the high-order generalization of Hermitian matrices. As the role played by Hermitian matrices in matrix theory and quadratic optimization, CPS tensors have shown growing interest recently in tensor theory and optimization, particularly in many application-driven complex polynomial optimization problems. In this paper, we study CPS tensors with a focus on ranks, rank-one decompositions and approximations, as well as their applications. The analysis is conducted along side with a more general class of complex tensors called partial-symmetric tensors. We prove constructively that any CPS tensor can be decomposed into a sum of rank-one CPS tensors, which provides an alternative definition of CPS tensors via linear combinations of rank-one CPS tensors. Three types of ranks for CPS tensors are defined and shown to be different in general. This leads to the invalidity of the conjugate version of Comon's conjecture. We then study rank-one approximations and matricizations of CPS tensors. By carefully unfolding CPS tensors to Hermitian matrices, rank-one equivalence can be preserved. This enables us to develop new convex optimization models and algorithms to compute best rank-one approximation of CPS tensors. Numerical experiments from various data are performed to justify the capability of our methods.
The goal of tensor completion is to fill in missing entries of a partially known tensor under a low-rank constraint. In this paper, we mainly study low rank third-order tensor completion problems by using Riemannian optimization methods on the smooth manifold. Here the tensor rank is defined to be a set of matrix ranks where the matrices are the slices of the transformed tensor obtained by applying the Fourier-related transformation onto the tubes of the original tensor. We show that with suitable incoherence conditions on the underlying low rank tensor, the proposed Riemannian optimization me…
A symmetric tensor, which has a symmetric nonnegative decomposition, is called a completely positive tensor. We consider the completely positive tensor decomposition problem. A semidefinite algorithm is presented for checking whether a symmetric tensor is completely positive. If it is not completely positive, a certificate for it can be obtained; if it is completely positive, a nonnegative decomposition can be obtained.
In this paper we study multivariate polynomial functions in complex variables and the corresponding associated symmetric tensor representations. The focus is on finding conditions under which such complex polynomials/tensors always take real values. We introduce the notion of symmetric conjugate forms and general conjugate forms, and present characteristic conditions for such complex polynomials to be real-valued. As applications of our results, we discuss the relation between nonnegative polynomials and sums of squares in the context of complex polynomials. Moreover, new notions of eigenvalue…
Hermitian tensors are natural generalizations of Hermitian matrices, while possessing rather different properties. A Hermitian tensor is separable if it has a Hermitian decomposition with only positive coefficients, i.e., it is a sum of rank-1 psd Hermitian tensors. This paper studies how to detect separability of Hermitian tensors. It is equivalent to the long-standing quantum separability problem in quantum physics, which asks to tell if a given quantum state is entangled or not. We formulate this as a truncated moment problem and then provide a semidefinite relaxation algorithm to solve it.…
Tensor decomposition has been widely used in machine learning and high-volume data analysis. However, large-scale tensor factorization often consumes huge memory and computing cost. Meanwhile, modernized computing hardware such as tensor processing units (TPU) and Tensor Core GPU has opened a new window of hardware-efficient computing via mixed- or low-precision arithmetic representations. In this paper, we exploit the low-precision representation of tensor factorization, and propose a mixed-precision block stochastic gradient descent (SGD) method to reduce the costs of CP tensor decomposition…
Alternating least squares (ALS) is often considered the workhorse algorithm for computing the rank-R canonical tensor approximation, but for certain problems its convergence can be very slow. The nonlinear conjugate gradient (NCG) method was recently proposed as an alternative to ALS, but the results indicated that NCG is usually not faster than ALS. To improve the convergence speed of NCG, we consider a nonlinearly preconditioned nonlinear conjugate gradient (PNCG) algorithm for computing the rank-R canonical tensor decomposition. Our approach uses ALS as a nonlinear preconditioner in the NCG…
In order to better understand micromechanical phenomena such as viscoelasticity and plasticity, the thermomechanical viewpoint is of prime importance but requires calorimetric measurements to be performed during a deformation process. Infrared imaging is commonly used to this aim but does not provide direct access to the intrinsic volumetric Thermomechanical Heat Sources (THS). An inversemethod is needed to convert temperature fields in the former quantity. The one proposed here relies on adiffusion-advection heat transfer model. Advection is generally not considered in such problems but due t…
In this paper we examine a symmetric tensor decomposition problem, the Gramian decomposition, posed as a rank minimization problem. We study the relaxation of the problem and consider cases when the relaxed solution is a solution to the original problem. In some instances of tensor rank and order, we prove generically that the solution to the relaxation will be optimal in the original. In other cases, we present interesting examples and approaches that demonstrate the intricacy of this problem.
Tensor Networks for Latent Variable Analysis. Part I: Algorithms for Tensor Train Decomposition
Brain Science Institute, RIKEN（RIKEN脑科学研究所）；Systems Research Institute PAS（波兰科学院系统研究所）；University of Bonn（波恩大学）；Institute of Information Theory and Automation（信息与自动化研究所）；Georgetown University（乔治城大学）；Imperial College（帝国理工学院）
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 the rank of a tensor. We determine the nuclear norm of various tensors of interest. Along the way, we also do a systematic study various measures of orthogonality in tensor product spaces and we give a new generalization of the Singular Value Decomposition to higher order tensors.
In general, algorithms for order-3 CANDECOMP/-PARAFAC (CP), also coined canonical polyadic decomposition (CPD), are easily to implement and can be extended to higher order CPD. Unfortunately, the algorithms become computationally demanding, and they are often not applicable to higher order and relatively large scale tensors. In this paper, by exploiting the uniqueness of CPD and the relation of a tensor in Kruskal form and its unfolded tensor, we propose a fast approach to deal with this problem. Instead of directly factorizing the high order data tensor, the method decomposes an unfolded tens…
