A concise proof to the spectral and nuclear norm bounds through tensor partitions

Abstract On estimations of the lower and upper bounds for the spectral and nuclear norm of a tensor, Li established neat bounds for the two norms based on regular tensor partitions, and proposed a conjecture for the same bounds to be hold based on general tensor partitions [Z. Li, Bounds on the spectral norm and the nuclear norm of a tensor based on tensor partition, SIAM J. Matrix Anal. Appl., 37 (2016), pp. 1440-1452]. Later, Chen and Li provided a solution to the conjecture [Chen B., Li Z., On the tensor spectral p-norm and its dual norm via partitions]. In this short paper, we present a concise and different proof for the validity of the conjecture, which also offers a new and simpler proof to the bounds of the spectral and nuclear norms established by Li for regular tensor partitions. Two numerical examples are provided to illustrate tightness of these bounds.


Introduction
Tensor is the main subject in multilinear algebra [1][2][3][4][5][6]. Let R be the eld of real numbers. Specially, a tensor T = (t i i ···i d ) ∈ R n ×n ×···×n d is a d-way array, i.e., its entries t i i ···i d are represented via d indices, say i , i , · · · , i d with each index ranging from to n j , ≤ j ≤ d. T is also called as an n × n × · · · × n d tensor. Similar to the de nition of the submatrix of a matrix, a p × p × · · · × p d subtensor of a tensor T ∈ R n ×n ×···×n d is a p × p × · · · × p d tensor formed by taking a block of the entries from the original tensor T.
Some general notations are in place: tensors are denoted by the calligraphic letters (e.g. T or X), scalars are denoted by plain letters, and matrices and vectors are denoted by bold letters (e.g. X and x).
Let · , · , and · ∞ denote the conventional -norm, -norm, and ∞-norm of a vector, respectively, i.e., De nition 1.1. Let T ∈ R n ×n ×···×n d . The spectral norm of T denoted by T σ is de ned as where ·, · is the classical Euclidean inner product, and the symbol " • " denotes the outer product operation of vectors such that the entries of De nition 1.2. Let T ∈ R n ×n ×···×n d . The nuclear norm of T denoted by T * is de ned as With the wide applications of the spectral and nuclear norm of a matrix, the research on the tensor spectral and nuclear norm has also attracted much attention recently. Unlike the computation of the spectral and nuclear norm of a matrix that can be done easily, the tensor spectral and nuclear norm are both NP-hard to compute; see [7] and [8]. Therefore, estimating these bounds, especially the polynomial-time approximation bounds has been a hot issue [7][8][9][10][11][12][13][14].
In [15], Li proposed an e cient way for the estimation of the tensor spectral and nuclear norms based on tensor partitions, which is de ned as follows.
De nition 1.3. [15] A partition {T , T , · · · , Tm} is called a general tensor partition of a tensor T ∈ R n ×n ×···×n d if every T j (j = , , · · · , m) is a subtensor of T, every pair of subtensors {T i , T j } with i ≠ j has no common entry of T, and every entry of T belongs to one of the subtensors in {T , T , · · · , Tm}.
Furthermore, let X ∈ R n ×n ×···×n d and {X , X , · · · , Xm} be a general tensor partition of X. If each X j has the same partition way as T j for ≤ j ≤ m, then T and X are said to have the same partition pattern.
Li also proposed a special tensor partition called regular tensor partition based on which the bounds of tensor norms were established [15]. The partition is obtained via several tensor cuts. We omit the details as it is not relevant to our discussion here. For illustration, Fig. 1 depicts a general tensor partition of a third order tensor. For a general tensor partition of a tensor, Li presented the following conjecture, which was answered in a rmative via a lengthy proof and also extended to a generalized tensor spectral and nuclear norms in a recent manuscript [11]. Some applications and general tightness results on rank-one tensors are discussed as well.
In the current paper, we, in an independent work , propose a much simpler way to prove this conjecture. We also provide some nontrivial examples to show the tightness of these bounds. Since a regular tensor partition is a special type of a general tensor partition, naturally, the way for the solution to the conjecture also o ers a new proof to the bounds for the spectral and nuclear norm established in [15]. The rest of this paper is organized as follows: In Section 2, we simply recall some de nitions and results required for the subsequent sections. In Section 3, the main results of the paper are presented. A short conclusion is given in Section 4.

Preliminaries
The main objective of this section is to review some basic de nitions and simple results relating to the tensor.

De nition 2.2. A tensor
Relating to the spectral norm of a tensor and the best rank-one approximation tensor, the following conclusion is straightforward.
Lemma 2.1. [8,16] Let T ∈ R n ×n ×···×n d . Suppose that u • u • · · · • u d is a best rank-one approximation to T, then We were not aware [11] in an earlier version of the current paper.
For the sake of convenience, we may write an n × n × n tensor T in the following form (T |T | · · · |Tn ), where T i ∈ R n ×n , ≤ i ≤ n . For example, let T = (t ijk ) ∈ R × × . Then T is expressed as the following form:

Main results
In this section, we provide a new proof to the Conjecture 1. Proof. Since W is a rank-one tensor, W can be written as where w j ∈ R n j , j = , , · · · , d. According to the de nition of the general partition, we know that every W i (i = , , · · · , m) is a subtensor of W. Without loss of generality, suppose that W i ∈ R n i, ×n i, ×···×n i,d , then it follows from (2) that W i can be written as the following form: where every w i,j ∈ R n i,j is a subvector of w j , i = , , · · · , m and j = , , · · · , d. This implies that every W i (i = , , · · · , m) is a rank-one tensor or a zero tensor. 2 We are ready to prove the Conjecture 1.1. For the sake of clarity, the Conjecture 1.1 is written as the Theorem 3.1.
Proof. Without loss of generality, we suppose that Based on the fact that the Frobenious norm of the best rank-one approximation to the subtenor of a tensor is less than or equal to the Frobenious norm of the best rank-one approximation to this tensor, the left hand side of inequality (3) is obviously true. Thus, we only need to prove the right hand side of inequality (3). Suppose that W ∈ R n ×n ×···×n d is a best rank-one approximation to T ∈ R n ×n ×···×n d . By Lemma 2.1, we get Furthermore, suppose that {W , W , · · · , Wm} is a general tensor partition of W with the same partition pattern as T, then it follows from the Lemma 2.1 that m j= Noting that every W j (j = , , · · · , m) is a rank-one tensor or a zero tensor (by Lemma 3.1), we get m j= Comparing (5) with (6), we get This implies that the right hand side of inequality (3) is true.
In what follows we will prove the inequality (4). Firstly, we prove the right hand side of the inequality (4). As mentioned in [15], the upper bound for the nuclear norm can be obtained through the de nition of the nuclear norm.
It follows from the de nition of the nuclear norm that Suppose that {X , X , · · · , Xm} is a general tensor partition of the arbitrary tensor X with the same partition pattern as T, then Secondly, we prove the left hand side of the inequality (4). It follows from the right hand side of inequality (3) that Then, according to the de nition of the nuclear norm of a tensor, we get Based on the arbitrariness of the tensor X, we can ensure that all its sub-tensors are non-zero tensors. Let X j σ = σ j , then σ j ≠ . Furthermore, let Y j = σ j X j , then Y j σ = and the inequality (7) can be written as It follows from (8) and (10) By applying the following general partition, the tensor T is partitioned into ve subtensors, each corresponding to one of the ve colors. Speci cally, let Then {T , T , T , T , T } is a general tensor partition of T. By a simple computation, we get and T * = .
Using the same method above, other upper bounds for the spectral norm and lower bounds for the nuclear norm can be obtained. For the sake of simplicity, we omit the corresponding discussions. Let is a rank-one tensor and Thus, the Frobenius norm of the best rank-one approximation to the tensor T is larger than or equal to Then it follows from (11) that This implies that for the tensor T in this simple example, the tensor partition {T , T , T , T , T } is the best choice of all tensor partitions for the estimation of the spectral norm of T. However, we do not know whether the lower bound of the nuclear norm, estimated by (12), is tight, since, there is no an e ective way for estimating the nuclear norm [8].
For the sake of completeness, in what follows, we give another example to illustrate that a tight lower bound of the nuclear norm can be obtained by the Theorem 3.1.
Furthermore, the tensor T can be decomposed into a sum of two rank-one tensors. That is Thus, it holds It follows from (13) and (14) that This implies that a tight lower bound of the nuclear norm is obtained.
However, neither the lower bound nor the upper bound given by (15) are tight. Actually, through a series of calculations, we get T σ = .
As discussed above, how to choose a better tensor partition for the estimation of the spectral norm and nuclear norm of a general tensor is no xed format, and it could be one of the future research.

Conclusions
In this paper, by considering the structure of the subtensors of rank-one tensors, we present a new proof to the conjecture proposed by Li [15]. The proof is di erent and simper than the method for proving the main results relating to the bounds for the spectral norm and nuclear norm in [11]. As discussed in [15], we believe these inequalities will have great potential in various applications. In the future, we will nd more applications of these inequalities.