Bounds for the Z-eigenpair of general nonnegative tensors

Abstract In this paper, we consider the Z-eigenpair of a tensor. A lower bound and an upper bound for the Z-spectral radius of a weakly symmetric nonnegative irreducible tensor are presented. Furthermore, upper bounds of Z-spectral radius of nonnegative tensors and general tensors are given. The proposed bounds improve some existing ones. Numerical examples are reported to show the effectiveness of the proposed bounds.


Introduction
We start with some preliminaries. First, denote OEn D f1; 2; ; ng. A real mth order n-dimensional tensor A D .a i 1 i 2 i m / consists of n m real entries: where i j D 1; 2; ; n for j 2 OEm [1][2][3][4][5]. It is obvious that a vector is an order 1 tensor and a matrix is an order 2 tensor. Moreover, a tensor A D .a i 1 i m / is called nonnegative (positive) if each entry is nonnegative (positive). A tensor A is said to be symmetric [6,7] if its entries a i 1 i 2 i m are invariant under any permutation of the indices. We shall denote the set of all real mth order n-dimensional tensors by R OEm;n , and the set of all nonnegative mth order n-dimensional tensors by R OEm;n C . For an n-dimensional vector x D .x 1 ; x 2 ; ; x n /, real or complex, we define the n-dimensional vector: ; and the n-dimensional vector: x OEm 1 WD x m 1 i Á 1ÄiÄn : The following two definitions were first introduced and studied by Qi and Lim [7,8].
Definition 1.1 ( [7,8]). Let A 2 R OEm;n . A pair . ; x/ 2 C .C n nf0g/ is called an eigenvalue-eigenvector (or simply eigenpair) of A if they satisfy the equation We call . ; x/ an H-eigenpair if they are both real. x T x D 1: We call . ; x/ a Z-eigenpair if they are both real.
The mth degree homogeneous polynomial of n variables f A .x/ associated with an mth order n-dimensional tensor A D .a i 1 i 2 i m / 2 R OEm;n can be represented as where x m can be regarded as an mth order n-dimensional rank-one tensor with entries x i 1 x i m [2,5,9], and Ax m is the inner product of A and x m . Following concept about weakly symmetric of tensors was first introduced and used by Chang, Pearson, and Zhang [6] for studying the properties of Z-eigenvalue of nonnegative tensor.
and the right-hand side is not identical to zero.
It should be noted for m D 2, symmetric matrices and weakly symmetric matrices are the same. However, it is shown in [6] that a symmetric tensor is necessarily weakly symmetric for m > 2, but the converse is not true in general. Thus, the results of this paper derived for weakly symmetric tensors, apply also for symmetric tensors.
In [8], the notion of irreducible tensors was introduced. otherwise, we say A is irreducible.
The Z-spectral radius of a tensor is defined as follows in [10]. , then there exists a Z-eigenvalue 0 0 and a nonnegative Z-eigenvector x 0 ¤ 0 of A such that Ax m 1 0 D 0 x 0 , in particular, if A is irreducible, then the eigenvalue 0 and the eigenvector x 0 are positive. Furthermore, if A 2 R OEm;n C is weakly symmetric irreducible, then the spectral radius %.A/ is a positive Z-eigenvalue with a positive Z-eigenvector.
Z-eigenvalues play a fundamental role in the symmetric best rank-one approximation which has numerous applications in engineering and higher order statistics, such as Statistical Data Analysis [2,5,9]. The symmetric best rank-one approximation of A D .a i 1 i 2 i m / is a rank-one tensor x m D . x i 1 x i 2 x i m /, where 2 R, x 2 R n , kxk 2 D 1 and kxk 2 is the Euclidean norm of x in R n , such that the Frobenius norm kA x m k F is minimized. The Frobenius norm of the tensor A D .a i 1 i 2 i m / has the form kAk F WD s X i 1 ;i 2 ;:::;i m 2OEn a 2 i 1 i 2 i m : According to [11], x m is a symmetric best rank-one approximation of A if and only if is a Z-eigenvalue of A with the largest absolute value, while x is a Z-eigenvector of A associated with the Z-eigenvalue . In particular, when A 2 R OEm;n C is weakly symmetric irreducible, %.A/x m 0 is a symmetric best rank-one approximation of A, where x 0 is a Z-eigenvector of A associated with Z-spectral radius %.A/, i.e., min 2R;x2R n ; Thus, we obtain the quotient of the residual of a symmetric best rank-one approximation of tensor A and the Frobenius norm of tensor A as follows: By Equalities (3) and (4), if we give a bound of Z-spectral radius of A, then a bound of min 2R;x2R n ;kxk 2 D1 kA x m k F and kA %.A/x m 0 k F kAk F will be obtained. It follows from [12][13][14][15][16] that the bound of gives a convergence rate for the greedy rank-one update algorithm.
Recently, some H-spectral of matrices have been successfully extended to higher order tensors [17][18][19]. For the Z-eigenpair case, Chang, Pearson and Zhang [10] discussed the variation principles of Z-eigenvalues of nonnegative tensors, as a corollary of the main results, they presented a lower bound of the Z-spectral radius for weakly symmetric nonnegative irreducible tensors as follows: For a general tensor case, they also provided an upper bound for the Z-spectral radius: Song and Qi [20] obtained the following upper bound for a general mth order n-dimensional tensor: He and Huang [21] gave a bound for a weakly symmetric positive tensor: Since Â.A/ Ä 1, it is easy to see that the bound (8) is smaller than those in (6) and (7) if the tensor is weakly symmetric positive. Recently, Li, Liu and Vong [22] have given a lower bound and an upper bound for a weakly symmetric nonnegative irreducible tensor: where and They also proved that the upper bound (10) is smaller than that in (8). Furthermore, Li, Liu and Vong [22] obtained the following upper bound for a general mth order n-dimensional tensor: In this paper, we continue this research on the Z-eigenpair and present some bounds as follows: for a weakly symmetric nonnegative irreducible tensor, we present a bound for Z-spectral radius, which improves the bound in (10). For a weakly symmetric nonnegative tensor, we give an upper bound for Z-spectral radius. Furthermore, for a nonnegative tensor and a general tensor, an upper bound for Z-spectral radius is also provided, which is tighter than the bound in (14) in some sense. Our paper is organized as follows. In Section 2, an upper bound for the ratio of the largest and smallest values of a Z-eigenvector is given. Also, a lower bound and an upper bound for the Z-spectral radius of a weakly symmetric nonnegative irreducible tensor are presented. Moreover, an upper bound for the Z-spectral radius of a weakly symmetric nonnegative tensor is provided. An upper bound for the Z-spectral radius of a nonnegative tensor and an upper bound for the Z-spectral radius of a general tensor are obtained in section 3. Numerical examples are presented in the final section.
We first add a comment on the notation that is used. For a tensor A, let jAj denote the tensor whose .i 1 ; ; i m /th entry is defined as ja i 1 i m j. For a set S, jS j denotes the number of elements of S . The function bxc indicates the integer round-down of x. Denote where j 2 OEn, k D 0; 1; ; m 1.

Bounds for weakly symmetric nonnegative tensors
In this section, a lower bound and an upper bound for the Z-spectral radius of a weakly symmetric nonnegative irreducible tensor are provided, which improves the bound (10). We first establish a lemma to estimate the ratio of the largest and smallest values of a Z-eigenvector.
is an irreducible tensor. Then for any Z-eigenpair . 0 ; x/ of A with a positive eigenvector x, we have x max where x mi n D min i 2OEn Proof.
For simplicity, let (17), we obtain Taking i D p in (18) and multiplying by x 1 p on the both sides of (18) gives Taking i D q. By (20) and the similar technique to (19) we have Combining (19) with (21) Note that x l x s Á m k 1 x l x s So the desired conclusion follows.
We next compare the bounds (15) in Lemma 2.1 with the corresponding bounds in Theorem 3.1 of [22], in which the authors presented the following bounds: where ı.A/ given as (11 We have from Equality (16)  which proves the first Inequality of (30). This proves the theorem. This implies the bounds (30) are always better than the corresponding bounds (10).
Remark 2.6. For an irreducible weakly symmetric tensor A D .a i 1 i 2 i m / 2 R OEm;n C , when m and n are very large, the bounds in (30) need more computations than the bounds in (10). As stated in Section 1, by the bounds in (30), we can obtain a more sharp bound of min 2R;x2R n ;kxk 2 D1 kA x m k F , which plays an important role in the symmetric best rank-one approximation [12][13][14][15][16]. This can be seen in the following example. Before generalizing the upper bound (30) of Theorem 2.4 to weakly symmetric nonnegative tensor, which will be used in Section 4, we first give a lemma in [20].  and E is a tensor with all entries being 1. Then fA t g is a sequence of weakly symmetric and positive tensors satisfying 0 Ä A < A t C1 < A t . By Lemma 2.8, %.A t / is a monotone decreasing sequence with lower bound %.A/ so that %.A t / has a limit. Thus, by Theorem 2.4, we have ; letting t ! 1, note that .A t / ! .A/, we obtain %.A/ Ä Á.A/. This yields the desired conclusion.

Upper bound for nonnegative tensors
In this section, we present an upper bound for the Z-eigenvalue of a nonnegative tensor and an upper bound for the Z-eigenvalue of a general tensor, which improves the bound (14). The following two lemmas will be used.  Based on Lemma 3.2, we have the following lemma.
Lemma 3.3. Suppose that A is weakly symmetric. Then Proof. Assume that A is a weakly symmetric tensor. By Lemma 3.2, we have On the other hand, assume that is a Z-eigenvalue of A with Z-eigenvector x. It follows from Equation (1.2) that D Ax m , thus which together with (37), yields (36). The proof is completed.
In the next theorem we have given an upper bound for the Z-eigenvalues of a nonnegative mth order n-dimensional tensor.  The proof is completed.
The following result gives an upper bound for the Z-eigenvalue of a general mth order n-dimensional tensor.  The proof is completed.
By means of the proof technique of Theorem 3.5, the following conclusions are obtained.
Corollary 3.6. Suppose that A D .a i 1 i 2 i m / 2 R OEm;n is a symmetric tensor. Then for any Z-eigenvalue , we have j j Ä %.jAj/ Ä Á.jAj/; where jAj D .ja i 1 i 2 i m j/ 2 R OEm;n C , and Á.jAj/ defined as Theorem 2.4.
Proof. Assume that is a Z-eigenvalue of A with Z-eigenvector x. It follows from Equation (1.2) that D Ax m , thus The proof is completed.

Comparisons with existing bounds
In this section, we will give some comparisons between our bounds and existing bounds for the Z-spectral radius for tensors. For a weakly symmetric nonnegative irreducible tensor, from Lemma 2.2, we obtain the bounds in (30), which are always better than the ones in (10). In particular, for a weakly symmetric positive tensor, the upper bound in (30) is always smaller than ones in (8).
The following example given in [10,22] shows the efficiency of the new bound (30). This example shows that the bound (30) is the best.
For a general tensor, the following example shows that our bound in (40) is better than the bound in (14) for some tensors.
Example 4.2. Randomly generate 1000 tensors of 4th order 3-dimensional tensor such that the elements of each tensor are generated by uniform distribution . 0:05; 0:40/. We compare the upper bounds of the Z-spectral radius of general tensor in (14) and (40). The numerical results are showed in Fig. 1. The x-axis of Fig. 1 refers to the sth random generated tensor. The plus symbol in blue color denotes the difference between the upper bound (14) and (40). As observed from Fig. 1, there are 96:2% of cases above the x-axis.