Jiawang Nie

Linghao Zhang, Ioana Dumitriu, Jiawang Nie

We study the rank one completion problem for tensors of arbitrary orders. The notion of rank one determinable tensors is introduced. We explore its properties and propose a recursive algorithm for computing rank one tensor completion. This algorithm only requires solving linear systems and computing singular vectors. In the absence of noise, it produces a unique rank one completion under some assumptions. In the presence of noise, we show that the computed rank one tensor completion is close to the exact one when the noise is sufficiently small. Numerical experiments demonstrate the efficiency and accuracy of the proposed method.

Jiawang Nie, Li Wang, Zequn Zheng

Multi-view learning is frequently used in data science. The pairwise correlation maximization is a classical approach for exploring the consensus of multiple views. Since the pairwise correlation is inherent for two views, the extensions to more views can be diversified and the intrinsic interconnections among views are generally lost. To address this issue, we propose to maximize higher order correlations. This can be formulated as a low rank approximation problem with the higher order correlation tensor of multi-view data. We use the generating polynomial method to solve the low rank approximation problem. Numerical results on real multi-view data demonstrate that this method consistently outperforms prior existing methods.

Jiawang Nie

For a given symmetric tensor, we aim at finding a new one whose symmetric rank is small and that is close to the given one. There exist linear relations among the entries of low rank symmetric tensors. Such linear relations can be expressed by polynomials, which are called generating polynomials. We propose a new approach for computing low rank approximations by using generating polynomials. First, we estimate a set of generating polynomials that are approximately satisfied by the given tensor. Second, we find approximate common zeros of these polynomials. Third, we use these zeros to construct low rank tensor approximations. If the symmetric tensor to be approximated is sufficiently close to a low rank one, we show that the computed low rank approximations are quasi-optimal.

Jiawang Nie

This paper studies symmetric tensor decompositions. For symmetric tensors, there exist linear relations of recursive patterns among their entries. Such a relation can be represented by a polynomial, which is called a generating polynomial. The homogenization of a generating polynomial belongs to the apolar ideal of the tensor. A symmetric tensor decomposition can be determined by a set of generating polynomials, which can be represented by a matrix. We call it a generating matrix. Generally, a symmetric tensor decomposition can be determined by a generating matrix satisfying certain conditions. We characterize the sets of such generating matrices and investigate their properties (e.g., the existence, dimensions, nondefectiveness). Using these properties, we propose methods for computing symmetric tensor decompositions. Extensive examples are shown to demonstrate the efficiency of proposed methods.

Jiawang Nie, Xinzhen Zhang

This paper discusses the computation of real Z-eigenvalues and H-eigenvalues of nonsymmetric tensors. A general nonsymmetric tensor has finitely many Z-eigenvalues, while there may be infinitely many ones for special tensors. In the contrast, every nonsymmetric tensor has finitely many H-eigenvalues. We propose Lasserre type semidefinite relaxation methods for computing such eigenvalues. For every nonsymmetric tensor that has finitely many real Z-eigenvalues, we can compute all of them; each of them can be computed by solving a finite sequence of semidefinite relaxations. For every nonsymmetric tensor, we can compute all its real H-eigenvalues; each of them can be computed by solving a finite sequence of semidefinite relaxations. Various examples are demonstrated.

Jiawang Nie

The low rank tensor approximation problem (LRTAP) is to find a tensor whose rank is small and that is close to a given one. This paper studies the LRTAP when the tensor to be approximated is close to a low rank one. Both symmetric and nonsymmetric tensors are discussed. We propose a new approach for solving the LRTAP. It consists of three major stages: i) Find a set of linear relations that are approximately satisfied by the tensor; such linear relations can be expressed by polynomials and can be found by solving linear least squares. ii) Compute a set of points that are approximately common zeros of the obtained polynomials; they can be found by computing Schur decompositions. iii) Construct a low rank approximating tensor from the obtained points; this can be done by solving linear least squares. Our main conclusion is that if the given tensor is sufficiently close to a low rank one, then the computed tensor is a good enough low rank approximation. This approach can also be applied to efficiently compute low rank tensor decompositions, especially for large scale tensors.

Chun-Feng Cui, Yu-Hong Dai, Jiawang Nie

This paper studies how to compute all real eigenvalues of a symmetric tensor. As is well known, the largest or smallest eigenvalue can be found by solving a polynomial optimization problem, while the other middle eigenvalues can not. We propose a new approach for computing all real eigenvalues sequentially, from the largest to the smallest. It uses Jacobian SDP relaxations in polynomial optimization. We show that each eigenvalue can be computed by solving a finite hierarchy of semidefinite relaxations. Numerical experiments are presented to show how to do this.

Simai He, Zhi-Quan Luo, Jiawang Nie et al.

In this paper we study the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) $\min \{x^* C x \mid x^* A_k x \ge 1, x\in\mathbb{F}^n, k=0,1,...,m\}$; and (2) $\max \{x^* C x \mid x^* A_k x \le 1, x\in\mathbb{F}^n, k=0,1,...,m\}$. If \emph{one} of $A_k$'s is indefinite while others and $C$ are positive semidefinite, we prove that the ratio between the optimal value of (1) and its SDP relaxation is upper bounded by $O(m^2)$ when $\mathbb{F}$ is the real line $\mathbb{R}$, and by $O(m)$ when $\mathbb{F}$ is the complex plane $\mathbb{C}$. This result is an extension of the recent work of Luo {\em et al.} \cite{LSTZ}. For (2), we show that the same ratio is bounded from below by $O(1/\log m)$ for both the real and complex case, whenever all but one of $A_k$'s are positive semidefinite while $C$ can be indefinite. This result improves the so-called approximate S-Lemma of Ben-Tal {\em et al.} \cite{BNR02}. We also consider (2) with multiple indefinite quadratic constraints and derive a general bound in terms of the problem data and the SDP solution. Throughout the paper, we present examples showing that all of our results are essentially tight.