Bounds for the Z-eigenpair of general nonnegative tensors

1 Introduction

We start with some preliminaries. First, denote [n] = {1, 2, …, n}. A real mth order n-dimensional tensor 𝓐 = (a_i1i2…_im) consists of n^m real entries:

where i_j = 1, 2, …, n for j ∈ [m] [1–5]. It is obvious that a vector is an order 1 tensor and a matrix is an order 2 tensor. Moreover, a tensor 𝓐 = (a_i1..._im) is called nonnegative (positive) if each entry is nonnegative (positive). A tensor 𝓐 is said to be symmetric [6, 7] if its entries a_i1i2…_im are invariant under any permutation of the indices. We shall denote the set of all real mth order n-dimensional tensors by ℝ^[m,n] and the set of all nonnegative mth order n-dimensional tensors by ℝ+[m,n]. For an n-dimensional vector x = .x₁, x₂, …, x_n), real or complex, we define the n-dimensional vector:

and the n-dimensional vector:

The following two definitions were first introduced and studied by Qi and Lim [7, 8].

Definition 1.1 ([7, 8]). Let 𝓐 ∈ ℝ^[m,n]. A pair (λ, x) ∈ ℂ× (ℂⁿ\{0}) is called an eigenvalue-eigenvector (or simply eigenpair) of A if they satisfy the equation

We call (λ, x) an H-eigenpair if they are both real.

Definition 1.2 ([7, 8]). Let 𝓐 ∈ ℝ^[m,n]. A pair (λ, x) ∈ ℂ × (ℂⁿ\{0}) is called an E-eigenvalue and E-eigenvector (or simply E-eigenpair) of 𝓐 if they satisfy the equation

We call (λ, 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 = (a_{i1i2 ... im}) ∈ ℝ^{[m, n]} can be represented as

where x^m can be regarded as an mth order n-dimensional rank-one tensor with entries x_i1 … x_im[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.

Definition 1.3 ([6]). A tensor 𝓐 = (a_i1i2…im) ∈ ℝ^{[m, n]}is called weakly symmetric, if the associated homogenous of polynomial f_𝓐(x) satisfy

and the right-hand side is not identical to zero.

It should be noted for m = 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.

Definition 1.4 ([8]). A tensor 𝓐 = (a_{i1i2 …im}) ∈ ℝ^{[m, n]}is called reducible if there exists a nonempty proper index subset I ⊂ [n], such that

otherwise, we say 𝓐 is irreducible.

The Z-spectral radius of a tensor is defined as follows in [10].

Definition 1.5 ([10]). Let 𝓐 ∈ ℝ^{[m, n]}. We define the Z-spectrum of 𝓐, denoted σ(𝓐) to be the set of all Z-eigenvalues of 𝓐. Assume σ(𝓐) ≠ ∅, then the Z-spectral radius of 𝓐, denoted ϱ(𝓐), is defined as

Chang, Pearson and Zhang [10] studied the Z-eigenpair problem for nonnegative tensors and presented the following Perron-Frobenius type theorem.

Lemma 1.6 ([10]). If𝓐∈ℝ+[m,n], then there exists a Z-eigenvalue λ₀ ≥ 0 and a nonnegative Z-eigenvector x₀ ≠ 0 of 𝓐 such that 𝓐x0m−1=λ0x0, in particular, if 𝓐 is irreducible, then the eigenvalue x₀and the eigenvector x₀are positive. Furthermore, if𝓐∈ℝ+[m,n]is weakly symmetric irreducible, then the spectral radius ϱ(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 = (a_i1i2…im) is a rank-one tensor νx^m = (νx_i1x_i2… x_im), where ν ∈ ℝ, x ∈ ℝⁿ, ||x||₂= 1 and ||x||₂ is the Euclidean norm of x in ℝⁿ, such that the Frobenius norm ||𝓐 – νx^m||_F is minimized. The Frobenius norm of the tensor 𝓐 = (a_i1i2…_im) has the form

According to [11], νx^m is a symmetric best rank-one approximation of 𝓐 if and only if ν is a Z-eigenvalue of 𝓐 with the largest absolute value, while x is a Z-eigenvector of 𝓐 associated with the Z-eigenvalue ν. In particular, when 𝓐∈ℝ+[m,n] is weakly symmetric irreducible, ρ(𝓐)x0m is a symmetric best rank-one approximation of 𝓐, where x₀ is a Z-eigenvector of 𝓐 associated with Z-spectral radius ϱ(𝓐), i.e.,

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 minν∈ℝ,x∈ℝ,||x||2=1||𝓐−νxm||F and ||𝓐−ρ(𝓐)x0m||F||𝓐||F will be obtained. It follows from [12-16] that the bound of ||𝓐−ρ(𝓐)x0m||F||𝓐||F 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 [17-19]. 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:

where ri(𝓐)=∑i2,...,im∈[n]aii2...im, for all i ∈ [n],

Since θ(𝓐) ≤ 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 |𝓐| denote the tensor whose .(i₁, … i_m)-th entry is defined as |a_{i1 … im}|. For a set S, |S| denotes the number of elements of S. The function └x┘ indicates the integer round-down of x. Denote

where j ∈ [n], k = 0, 1, …, m – 1.

2 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.

Lemma 2.1. Suppose that𝓐=(ai1i2...im)∈ℝ+[m,n]is an irreducible tensor. Then for any Z-eigenpair (λ₀, x) of 𝓐 with a positive eigenvector x, we have

wherexmin=mini∈[n]xi,xmax=maxi∈[n]xi,

Proof. Since (λ₀, x) is a Z-eigenpair of 𝓐 with x being positive, then

For simplicity, let x_l = x_max, x_s = x_min, rp(𝓐)=maxi∈[n]ri(𝓐)=R(𝓐),rq(𝓐)=mini∈[n]ri(𝓐)=r(𝓐). Consider the ith equation of (17), we obtain

Taking i = p in (18) and multiplying by xp−1 on the both sides of (18) gives

Similarly, we have

Taking i = q. By (20) and the similar technique to (19) we have

Combining (19) with (21) together gives

Multiplying xlxsm−1 on the both sides of the above inequality yields

Note that (xlxs)m−k−1≥(xlxs)k+1, for all k=0,1,…,⌊m2⌋−1. Hence, by (22), we have

i.e. xlxs≥α(𝓐)1m, which together with (22) yields

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 δ(𝓐) given as (11).

Lemma 2.2. Suppose that 𝓐 = (a_{i1i2 …im}) ∈ ℝ^{[m, n]}is an irreducible tensor. Then

and

Proof. It follows from Inequality (3.10) of Li, Liu and Vong [22] that δ(𝓐) ≥ y(𝓐) ≥ 1. Hence, we only prove

and

We have from Equality (16) that

Note that α(𝓐)m−k−1m≥α(𝓐)k+1m, for all k=0,1,…,⌊m2⌋−1, and

hence

which together with (28), yields Inequality (26).

From Equality (16), we obtain

Since α(𝓐). ≥ γ(𝓐) ≥ 1 and m ≥ 2,

which together with Inequality (29), yields

which implies Inequality (27) holds. The proof is completed. □

Remark 2.3. If 𝓐 is a nonnegative irreducible tensor, Inequality(25)implies the bounds(15)are always larger than the bounds(23).

Now we establish an upper and a lower bound for Z-spectral radius of a weakly symmetric nonnegative irreducible tensor.

Theorem 2.4. Suppose that𝓐=(ai1i2...im)∈ℝ+[m,n]is an irreducible weakly symmetric tensor. Then

where

and τ(𝓐) is given by Lemma 2.1.

Proof. It follows from Lemma 1.6 that there exists a positive Z-eigenvector x corresponding to ϱ(𝓐). Taking the similar technique of (18) we obtain

Since x^Tx = 1, we have xim−1≤xi, which together with (32) yields

Taking i = l and multiplying xl1−m on the both sides of the above inequality gives

This proves the second Inequality (30). On the other hand, it is known from Theorem 3.3 of Li, Liu and Vong [22] that for a weakly symmetric nonnegative irreducible tensor 𝓐, we have

which together with (15) yields

which proves the first Inequality of (30). This proves the theorem. □

Remark 2.5. It follows from Inequality(25)that for the parameters η(𝓐) and μ(𝓐) given by(31)we have

and

This implies the bounds(30)are always better than the corresponding bounds(10).

Remark 2.6. For an irreducible weakly symmetric tensor𝓐=(ai1i2...im)∈ℝ+[m,n], 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 ofminν∈ℝ,x∈ℝ,||x||2=1||𝓐−νxm||F, which plays an important role in the symmetric best rank-one approximation [12-16]. This can be seen in the following example.

Example 2.7. Let𝓐=(ai1i2i3i4)∈ℝ+[4,2]with entries defined as follows:

It is not difficult see that 𝓐 is a weakly symmetric nonnegative irreducible tensor, and

By the bound(10), we have

which implies