Advances in Numerical Analysis

Volume 2014 (2014), Article ID 109525, 10 pages

http://dx.doi.org/10.1155/2014/109525

Research Article

The Perturbation Bound for the Spectral Radius of a Nonnegative Tensor

Wen Li¹ and Michael K. Ng²

¹School of Mathematical Sciences, South China Normal University, Guangzhou, China

²Department of Mathematics, Hong Kong Baptist University, Hong Kong

Received 29 March 2014; Revised 29 August 2014; Accepted 9 September 2014; Published 28 September 2014

Academic Editor: Nils Henrik Risebro

Copyright © 2014 Wen Li and Michael K. Ng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the perturbation bound for the spectral radius of an mth-order n-dimensional nonnegative tensor . The main contribution of this paper is to show that when is perturbed to a nonnegative tensor by , the absolute difference between the spectral radii of and is bounded by the largest magnitude of the ratio of the ith component of and the ith component , where is an eigenvector associated with the largest eigenvalue of in magnitude and its entries are positive. We further derive the bound in terms of the entries of only when is not known in advance. Based on the perturbation analysis, we make use of the NQZ algorithm to estimate the spectral radius of a nonnegative tensor in general. On the other hand, we study the backward error matrix and obtain its smallest error bound for its perturbed largest eigenvalue and associated eigenvector of an irreducible nonnegative tensor. Based on the backward error analysis, we can estimate the stability of computation of the largest eigenvalue of an irreducible nonnegative tensor by the NQZ algorithm. Numerical examples are presented to illustrate the theoretical results of our perturbation analysis.

1. Introduction

Let be the real field. We consider an th order -dimensional tensor consisting of entries in :

is called nonnegative (or, resp., positive) if (or, resp., ). In the following discussion, for a given vector , we define a tensor-vector multiplication to be -dimensional column vector with its th entries given by

Also we let be the positive cone, and we let the interior of be denoted as . When (or ), is a nonnegative (or a positive) vector. We denote the th component of by or , and we denote the th entry of by or .

In recent studies of numerical multilinear algebra, eigenvalue problems for tensors have brought special attention. There are many related applications in information retrieval and data mining; see, for instance, [1, 2].

Definition 1. Let be an th-order -dimensional tensor and let be the set of all complex numbers. Assume that is not identical to zero. We say is an -eigenpair of if

Here, . The spectral radius of is defined by the largest eigenvalue of in magnitude.

This definition was introduced by Qi [3] when is even and is symmetric. Independently, Lim [4] gave such a definition but restricted to be a real vector and to be a real number. For the largest -eigenvalue of a nonnegative tensor, the Perron-Frobenius theorem was proved by Chang et al. [5], Friedland et al. [6], and Lim [4]. Y. Yang and Q. Yang [7] generalized the weak Perron-Frobenius theorem to general nonnegative tensors. In [4], the concept of irreducibility in nonnegative matrices has been extended to nonnegative tensors.

Definition 2. An th-order -dimensional tensor is called reducible if there exists a nonempty proper index subset such that

If is not reducible, then we call irreducible.

Theorem 3. Suppose is an th-order -dimensional nonnegative tensor.

(i)Then there exist and such that

(ii)If is irreducible, then and . Moreover, if is an eigenvalue with a nonnegative eigenvector, then . If is an eigenvalue of , then .

We call a Perron vector of a nonnegative tensor corresponding to its largest nonnegative eigenvalue.

Some algorithms for computing the largest eigenvalue of an irreducible tensor were proposed; see, for instance, [8–10]. However, the perturbation analysis and the backward error analysis for these algorithms have not been studied, which are important to the analysis of the accuracy and stability for computing the largest eigenvalue by these algorithms.

In this paper, we are interested in studying the perturbation bound for the spectral radius of an th-order -dimensional nonnegative tensor . The main contribution of this paper is to show that when is perturbed to a nonnegative tensor by and has a positive Perron vector , we have

where is a diagonal matrix, for an th-order -dimensional tensor and an -by- matrix , and the tensor-matrix multiplication [11] is a tensor of order and dimension with its entries given by

the maximum norm of a tensor is defined as follows:

The perturbation bound (6) shows that the absolute difference between the spectral radii of and is bounded by the largest magnitude of the ratio of the th component of and the th component . We further derive the bound based on the entries of when is not necessary to be known. Moreover, there is no convergence result of numerical algorithms [9, 10] for computing the spectral radius of a nonnegative tensor in general. We will make use of our perturbation results to estimate the spectral radius of a nonnegative tensor in general via the NQZ algorithm (see [10]).

On the other hand, we will study the backward error matrix and obtain its smallest error bound of for

such that is an irreducible nonnegative tensor, is the largest eigenvalue of , and is a Perron vector of by the NQZ algorithm. Our theoretical results show that can be chosen as follows:

where and denotes the vector 2-norm. By using these backward error results, we will evaluate the stability of computation of the largest eigenvalue of an irreducible nonnegative tensor by the NQZ algorithm.

The paper is organized as follows. In Section 2, we review the existing results and present the results for the perturbation bound of the spectral radius of a nonnegative tensor. In Section 3, we give the explicit expression of the backward error for the computation of the largest eigenvalue of an irreducible nonnegative tensor by the NQZ algorithm. Finally, concluding remarks are given in Section 4.

2. The Perturbation Bound of the Spectral Radius

Let us first state some preliminary results of a nonnegative tensor.

Lemma 4 (see [7, Lemma 5.2]). Suppose is an th-order -dimensional nonnegative tensor. Then we have

Lemma 5 (see [7, Lemma 5.1]). Suppose is an th-order -dimensional nonnegative tensor. If

then .

Lemma 6. Suppose is an th-order -dimensional nonnegative tensor with a positive Perron vector . Then one has

where .

Proof. By Theorem 3, we note that and is the largest eigenvalue of . Therefore,

The result follows.

Theorem 7. Suppose is an th-order -dimensional nonnegative tensor, is the perturbed nonnegative tensor of , and has a positive Perron vector . Then one has

where and is a vector of all ones.

Proof. Let us consider the tensor , where is the identity tensor.

For an th-order -dimensional identity tensor ,

It is clear that the eigenvalue of is the same as the eigenvalue of for any positive diagonal matrix . Then we obtain

By choosing , where is a positive Perron vector of , we have, for ,

On the other hand, we can deduce that, for ,

Taking such that

and then by using (20) with , we give the combination of (18) and (19) as follows:

Since the eigenvalue of is the same as the eigenvalue of , by using Lemma 4, (21) becomes

The right hand side of (15) is established. By using the above argument, we can show the left hand side of (15).

By Theorem 7, it is easy to obtain the following corollary.

Corollary 8. Suppose is an th-order -dimensional nonnegative tensor, is the perturbed nonnegative tensor of , and has a positive Perron vector . Then one has

where .

It is noted that can also be written as the perturbed tensor of ; that is, . Then, by Theorem 7, we have the following bound.

Corollary 9. Suppose is an th-order -dimensional nonnegative tensor with a positive Perron vector , and is the perturbed nonnegative tensor of . Then one has

where .

According to Corollary 9, we know that the absolute difference between the spectral radii of and is bounded by the largest magnitude of the ratio of the th component of and the th component .

Remark 10. In the nonnegative matrix case (), if is irreducible nonnegative matrix with positive Perron vector , then the perturbation of the eigenvalues of and is given by

see [12]. It is easy to see that our perturbation result in Corollary 9 can be reduced to the bound in (25).

Remark 11. In [6], it has been shown that if is symmetric and weakly irreducible, then has a positive Perron vector. In particular, when is irreducible, also has a positive Perron vector; see [4–6].

In Corollary 9, a Perron vector must be known in advance so that the perturbation bound can be computed. Here we derive the perturbation bound in terms of the entries of only.

Lemma 12. Suppose is an th-order () -dimensional positive tensor. Then one has

where is the positive Perron vector, , and .

Proof. Since is a positive tensor, must be irreducible and has a positive Perron vector. The left hand side of (26) is straightforward. Now we consider

for . It implies that

Similarly, we get

Thus we obtain

We note that, for any positive numbers , , and (), we have

As is a positive tensor and is a positive vector, it follows from the above inequalities and (30) that

Because the above upper bound is valid for , , we take the minimum among them over , and we obtain the following inequality:

The result follows.

Based on Lemma 12, we have the following lemma.

Lemma 13. Suppose is an th-order -dimensional positive tensor (), and is the perturbed nonnegative tensor of . Then one has

where

Proof. By using (25) in Corollary 9, we have

The result follows.

Remark 14. The perturbation bound in Lemma 13 can be achieved by considering and , where is a positive real number and is a tensor with all the entries being equal to one. It is clear that is irreducible; that is, has the unique positive Perron vector. And we just note that the largest eigenvalues of and are equal to and , respectively, , and .

Corollary 15. Suppose is an th-order -dimensional nonnegative tensor, and is the perturbed positive tensor of . Then one has

where is defined in (35).

Proof. We just switch the roles of and in Lemma 13.

Let us state the following lemma to handle the case when is a nonnegative tensor.

Lemma 16. Suppose is an th-order -dimensional nonnegative tensor, and . Then .

The proof of this lemma is similar to Theorem 2.3 given in [7].

Suppose that is a nonnegative tensor. Let . It is clear that is a positive tensor. Let be a perturbed nonnegative tensor of . By applying Corollary 15 to and , we have

Therefore, when , by Lemma 16, we have the following theorem.

Theorem 17. Suppose is an th-order -dimensional nonnegative tensor, and is the perturbed nonnegative tensor of . Then one has

provided that .

Example 18. We conduct an experiment to verify the perturbation bound. We randomly construct a positive tensor , where each entry is generated by uniform distribution . We further check the value of each entry must be greater than zero, and therefore the constructed tensor is positive. In the experiment, is perturbed to a positive tensor by adding , where is a positive number and is a positive tensor randomly generated by the above-mentioned method. We study the absolute difference between the spectral radii of and and the perturbation bound in Corollary 9. In Figure 1, we show the results for and . For each point in the figure, we give the average value of , , or based on the computed results for 100 randomly constructed positive tensors. The -axis refers to values of : 0.01, 0.005, 0.001, 0.0005, 0.0001, and 0.00005. We see from the figures that the average values (in logarithm scale) depend linearly on (in logarithm scale). The perturbation bounds and provide the upper bound of . This result is consistent with our prediction in the theory. It is interesting to note that the bound is very close to the actual difference.

(a)

(b)

Figure 1: The results of the perturbation bounds (a) and and (b) and : — (the actual difference); – – – – and – · – · – · – · .

2.1. Application to the NQZ Algorithm

In this subsection, we apply our perturbation results to the NQZ algorithm [10] which is an iterative method for finding the spectral radius of a nonnegative tensor. The NQZ algorithm presented in [10] is given as follows.

Choose and let . For , compute

It is shown in [10] that the sequences and converge to some numbers and , respectively, and we have . If , the gap is zero and therefore both the sequences and converge to . However, a positive gap may happen, which can be seen in an example given in [10]. Pearson [13] introduced the notion of essentially positive tensors and showed the convergence of the NQZ algorithm for essentially positive tensors. Chang et al. [8] further established the convergence of the NQZ algorithm for primitive tensors.

Definition 19. An th-order -dimensional tensor is essentially positive if for any nonzero .

Definition 20. An th-order -dimensional tensor is called primitive if there exists a positive integer such that for any nonzero .

An essentially positive tensor is a primitive tensor, and a primitive tensor is an irreducible nonnegative tensor but not vice versa.

Now we demonstrate how to use the results in Theorem 7 to estimate the spectral radius of a nonnegative tensor via the NQZ algorithm. For example, when is a reducible nonnegative tensor, then the NQZ algorithm may not be convergent. Our idea is to take a small perturbation , where is a very small positive number and is a tensor with all the entries being equal to one. It is clear that is essentially positive. Therefore, when we apply the NQZ algorithm to compute the largest eigenvalue of , it is convergent. Without loss of generality, we normalize the output vector in the NQZ algorithm. In particular, the 1-norm of is employed. The output of the algorithm contains a positive number and with (i.e., ). We note that

Then, by Theorem 7, we have

That is,

Let and . By considering , we know that

that is, we have

or

On the other hand,

By putting (46) and (47) into (43), we have

One may know if

This shows that we can estimate of a nonnegative tensor with a specified precision via the NQZ algorithm for the computation of .

Remark 21. Liu et al. [9] modified the NQZ algorithm for computing the spectral radius of , where is an irreducible nonnegative tensor and is a very small number, and showed that the algorithm converges to . This fact can be explained by our theoretical analysis in Theorem 7 by setting . We just note that the left and right sides of the inequality in (15) are equal to ; that is, . It implies that . However, when the given nonnegative tensor is not irreducible, then their results cannot be valid. However, our approach can still be used to estimate the spectral radius of a nonnegative tensor in general.

Example 22. In [8], the following 3rd order 3-dimensional nonnegative tensor is considered: and the other entries are equal to zero. It can be shown that this tensor is irreducible, but not primitive. There is no convergence result of the NQZ algorithm for such , and the spectral radius of is equal to ; see [8]. In this example, we take the perturbation . We apply the NQZ algorithm to compute the spectral radius of to approximate the actual one. According to Table 1, the results show that is about .

Table 1: Numerical results for different in Example 22.

Difference

1.399817488643705
1.428757688931172
0.028940200287467

1.412729187546902
1.415699496853463
0.002970309306561

1.414064662464100
1.414362477963432
0.000297815499332

1.414198667753479
1.414228457171375
0.000029789417896

1.414212073004730
1.414215052025221
0.000002979020491

Example 23. We consider the following 3rd order 3-dimensional nonnegative tensor: and the other entries are equal to zero. It can be shown that this tensor is reducible. The spectral radius of is equal to 3. There is no convergence result of the NQZ algorithm for such reducible nonnegative tensor. It is interesting to note that this tensor satisfies (49). In this example, we take the perturbation . The spectral radii of by the NQZ algorithm are still accurate approximations of : 3.090000000000000 (), 3.009000000000000 (), 3.000900000000000 (), 3.000090000000001 (), and 3.000009000000000 (). Indeed, both the values in the left and right sides of (43) are equal to 3.000000000000000 (up to the number of decimals shown in MATLAB). These results show the bounds are very tight and is about .

Example 24. Now we consider the following 3rd order 3-dimensional nonnegative tensor: and the other entries are equal to zero. is a reducible nonnegative tensor. For such , we know the is the spectral radius of and is the corresponding eigenvector (the normalized in 2-norm). In this example, we take the perturbation . Although this tensor does not satisfy (49), the spectral radii of by the NQZ algorithm are still accurate approximate of : 1.015565072567277 (), 1.001139501996208 (), 1.000104123060726 (), 1.000010127700654 (), and 1.000001004012030 (). We find that the values in the left and right sides of (43) are close to 1 and 0, respectively, for different values of . It is clear that the bounds are not tight. It is interesting to note that the maximum and minimum values of the entries of the associated eigenvector are 1 and 0; we expect that the error bound in (43) can be very poor. Indeed, the errors between the actual eigenvector and the approximate eigenvector are 0.243064619277186 (), 0.077415571882820 (), 0.024493622285564 (), 0.007745927468297 (), and 0.002449488513126 (). Thus the approximation is more accurate for the largest eigenvalue than for the associated eigenvector.

3. Backward Error Bound

In this section, we study the backward error tensor and obtain its smallest error bound of such that

where and are irreducible nonnegative tensors with for , is the largest eigenvalue of , and is a Perron vector of . The backward error for the eigenpair is defined as follows:

where

and .

Before we show the results of the backward error in the computation of the largest eigenvalue of an irreducible nonnegative tensor, we need the following lemma given in [14, Theorem 9.1].

Lemma 25. Let , , and . Let

Then

if and only if , and when , , where denotes the conjugate transpose of a matrix, is the Moore-Penrose generalized inverse of , , and .

Now we show the main result of this section.

Theorem 26. Suppose is an th-order -dimensional irreducible nonnegative tensor. Let and be the output of the NQZ algorithm (stated in Section 2.1) applied to , and . Then the backward error for and of the perturbed irreducible nonnegative tensor is given by

and can be chosen as follows:

Proof. We note that

In order to analyze , we rewrite the above equation into a linear system as follows:

where is an -by- matrix with its entries given by

It follows from Lemma 25 that

where is an -by- vector given by and is an -by- matrix. Therefore, we obtain

The minimization of is equivalent to finding a matrix such that can be minimized. It is clear that we choose ; that is, .

According to the NQZ algorithm stated in Section 2.1, we know that is nonnegative. As is positive, both and are also positive. Let us consider two cases. We note that if is a zero vector, then is the largest eigenvalue of , and the problem is solved. Now we consider is not a zero vector. It follows that at least one row of must be positive. It can be shown that is irreducible. The argument is that for any nonempty proper index subset such that

Hence the result follows.

Remark 27. When , the results of the backward error are reduced to the results in Theorem 6.3.2 in [14]; that is,

Example 28. Let us conduct an experiment to check the backward error when we use the NQZ algorithm for computing the largest eigenvalue of an irreducible nonnegative tensor. We randomly construct a positive tensor , where each entry is generated by uniform distribution . We further check the value of each entry must be greater than zero, and therefore the constructed tensor is positive. In Table 2, we show the average backward error based on the computed results for 100 randomly constructed positive tensors. We see from the table that the backward error can be very small, but it still depends on the order of and the dimension of the tensor. The backward error is large when or is large.

Table 2: The backward errors for different cases.

Backward error

3
5

3
10

3
20

3
40

4
5

4
10

4
20

4
40

4. Concluding Remarks

In summary, we have studied and derived the perturbation bounds for the spectral radius of a nonnegative tensor. Numerical examples have been given to demonstrate our theoretical results. Also we have investigated the backward error matrix and obtain its smallest error bound for its perturbed largest eigenvalue and associated eigenvector of an irreducible nonnegative tensor. Numerical examples have shown that the backward error can be very small. These results may show that the NQZ algorithm may be backward stable for the computation of the largest eigenvalue of an irreducible nonnegative tensor.

In this paper, we do not study the perturbation bound of the eigenvector corresponding to the largest eigenvalue of a nonnegative tensor. This would be an interesting research topic for future consideration.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referee for his/her valuable suggestions. Wen Li was supported in part by National Natural Science Foundations of China (nos. 10971075, 11271144), Guangdong Provincial Natural Science Foundations (no. s2012010009985), and Project of Department of Education of Guangdong Province (no. 2013KJCX0053). Research was supported in part by RGC GRF Grant no. 201812.

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