A Tensor Rank Theory and Maximum Full Rank Subtensors

04/22/2020 ∙ by Liqun Qi, et al. ∙ South China Normal University Tianjin University 0

A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. This property is one of the corner stones of the matrix rank theory. We call this property the max-full-rank-submatrix property. Tensor ranks play a crucial role in low rank tensor approximation, tensor completion and tensor recovery. However, their theory is still not matured yet. Can we set an axiom system for tensor ranks? Can we extend the max-full-rank-submatrix property to tensors? We explore these in this paper. We first propose some axioms for tensor rank functions. Then we introduce proper tensor rank functions. The CP rank is a tensor rank function, but is not proper. There are two proper tensor rank functions, the max-Tucker rank and the submax-Tucker rank, which are associated with the Tucker decomposition. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. We introduce the full rank tensor concept, and define the max-full-rank-subtensor property. We show the max-Tucker tensor rank function and the smallest tensor rank function have this property. We define the closure for an arbitrary proper tensor rank function, and show that it is still a proper tensor rank function and has the max-full-rank-subtensor property. An application of the submax-Tucker rank is also presented.

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1 Introduction

A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. We call this property the max-full-rank-submatrix property. This property is one of the corner stones of the matrix rank theory.

We now arrive the era of big data and tensors. Tensor ranks play a crucial role in low rank tensor approximation, tensor completion and tensor recovery [1, 4, 5, 6, 8, 10, 12, 13, 14, 15, 16, 17]. However, their theory is still not matured yet. Can we set an axiom system for tensor ranks? Can we extend the full rank concept and the max-full-rank-submatrix property to tensors? We explore these in this paper.

We first propose some axioms for tensor rank functions. Then we introduce proper tensor rank functions. The CP rank is a tensor rank function, but is not proper. There are two proper tensor rank functions, the max-Tucker rank and the submax-Tucker rank, which are associated with the Tucker decomposition. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. We introduce the full rank tensor concept, and define the max-full-rank-subtensor property. We show the max-Tucker tensor rank function and the smallest tensor rank function have this property. We define the closure for an arbitrary proper tensor rank function, and show that it is still a proper tensor rank function and has the max-full-rank-subtensor property. An application of the submax-Tucker rank is also presented.

The set of all nonnegative integers is denoted by . The set of all positive integers is denoted by . Let . Denote the set of all real th order tensors of dimension by . If , then we denote it by . Here “CT” means cubic tensors. Denote the set of all real tensors by

. Thus, scalars, vectors, matrices are a part of

. Let . We call a rank-one tensor if and only if there are nonzero vectors for , such that

Here, is the tensor outer product. Then, nonzero vectors and scalars are all rank-one tensors in this sense.

Suppose that and . Let and for . Suppose that for , with if for . Then we say that is a subtensor of . If , then we say that is a proper subtensor of . For , the subtensor , described above, is called a -row of if and for . If , then the corresponding -row is called the th -row of . The -row concept extends the concepts of rows and columns from matrices to tensors. For a matrix, a -row is called a row, a -row is called a column.

In the next section, we present a set of axioms for tensor rank functions. We list six properties which are essential for tensor ranks. In particular, we define a partial order “” among tensor rank functions, and show that there exists a unique smallest tensor rank function . We also introduce proper and strongly proper tensor rank functions in that section.

We study the CP rank and the Tucker rank in Section 3. The Tucker rank is a vector rank. We derive two scalar ranks from this, and call them the max-Tucker rank and the submax-Tucker rank respectively. We show that the CP rank, the max-Tucker rank and the submax-Tucker rank are all tensor rank functions. The CP rank is subadditive but not proper. The max-Tucker rank is proper, subadditive but not strongly proper. The submax-Tucker rank is strongly proper but not subadditive.

We introduce the concept of maximum full rank subtensors, and define the max-full-rank-subtensor property in Section 4. We show that the max-Tucker rank function has the max-full-rank-subtensor property. Suppose that and , and is a maximum full rank subtensor of under the max-Tucker rank. Then we show that there is an index , , such that all the th -rows of , with in the mode index set of , are linearly independent, and any -row of is a linear combination of the th -rows of with .

In Section 5, we define the closure of an arbitrary proper tensor rank function, and show that it is still a proper tensor rank function, and has the max-full-rank-subtensor property. We show that is strongly proper and has the max-full-rank-subtensor property.

We present an application of the submax-Tucker rank in internet traffic data approximation in Section 6.

Some final remarks are made in Section 7.

We use small letters to denote scalars, small bold letters to denote vectors, capital letters to denote matrices, and calligraphic letters to denote tensors. We denote the matrix rank of a matrix as .

2 Axioms and Properties of Tensor Rank Functions

Let . Consider . Suppose . An entry is called a diagonal entry of if . Otherwise, is called an off-diagonal entry of . If all the off-diagonal entries of are zero, then is called a diagonal tensor. If is diagonal, and all the diagonal entries of is , then is called the identity tensor of , and denoted as . Clearly, the identity tensor is unique to . The identity tensor plays an important role in spectral theory of tensors [7].

Definition 2.1

Suppose that . If satisfies the following six properties, then is called a tensor rank function.

Property 1 Suppose that . Then if and only if is a zero tensor, and if and only if is a rank-one tensor.

Property 2 For with , .

Property 3 Let , . Then is equal to the matrix rank of the matrix corresponding to .

Property 4 Let , , and is a real nonzero number. Then .

Property 5 Let , , and is a permutation on . Then , where , .

Property 6 Let . Suppose that , and is a subtensor of . Then .

These six properties are essential for tensor ranks. Property 1 specifies rank zero tensors and rank-one tensors. Though the tensor rank theory is not matured, there are no arguments in rank zero and rank-one tensors in the literature. Property 2 fixes the value of the tensor rank for identity tensors. This is necessary as identity tensors are good references for the magnitude of tensor ranks. Property 3 justifies the tensor rank is an extension of the matrix rank. Property 4 claims that the tensor rank is not changed when a tensor is multiplied by a nonzero real number. Property 5 says that the roles of the modes are balanced. Property 6 justifies the subtensor rank relation.

Suppose that are two tensor rank functions. If for any we always have , then we say that the tensor rank function is not greater than the tensor rank function and denote this relation as .

Theorem 2.2

Suppose that are two tensor rank functions. Define by

for any . Then is a tensor rank function, and .

Proof For any , let . Then Properties 1, 2, 3 and 4 hold clearly from the definition of tensor rank functions.

To show Property 5, we assume that is a permutated tensor of . Then and . Hence, and Property 5 is obtained.

Now we assume that is a subtensor of . Then and . Hence since and . Thus, Property 6 holds.

Thus, we conclude that is a tensor rank function.

clearly, and .

.

Theorem 2.3

There exists a unique tensor rank function , such that for any tensor rank function , we have .

Proof For any , define . This is well-defined as tensor rank functions take values on . Now we show that is a tensor rank function.

1) Suppose is a zero tensor in . Then for any tensor rank function , . This implies that by the definition of . On the other hand, suppose that for some . Then for some tensor rank function , . Hence, is a zero tensor from Property 1 of the tensor rank function . Similarly, we may show that if and only if is a rank-one tensor.

2) For any , for all tensor rank functions . Thus .

3) Let . Let be the corresponding matrix in . Then for any tensor rank function , . Hence all of are equal. Hence, and Property 3 holds.

4) For any and any tensor rank function , for any . Thus, .

5) We have Properties 5 and 6 in a similar way as in the proof of Theorem 2.2 and omit the details here.

By the definition, for any tensor rank function .

Suppose that and are two tensor rank functions with the property that and for any tensor rank function. Then . We see that . Thus, such a tensor rank function is unique.

.

We call the smallest tensor rank function. In Section 5, we will show that has the sub-full-rank property.

The six properties in Definition 2.1 are essential to tensor rank functions. There are some other properties which are satisfied by some tensor rank functions.

Definition 2.4

Suppose that is a tensor rank function. We say that is a proper tensor rank function if for any and , we have .

For an square matrix, its matrix rank is never greater than its dimension . Thus, proper tensor rank functions are reasonable in a certain sense.

Definition 2.5

Suppose that is a tensor rank function. We say that is a subadditive tensor rank function if for any , and , we have

The subadditivity property is somehow restrictive. The minimum of two subadditive tensor rank functions may not be subadditive.

Proposition 2.6

Suppose that is a proper tensor rank function. Let with and . Then we have

(2.1)

Proof Let and with a subtensor . Then from Property 6. Together with since is proper, the result is arrived.

.

The matrix rank of an rectangular matrix is never greater than . From this proposition, we may think further to restrict the magnitude of the tensor rank. For with , we define submax as the second largest value of .

Definition 2.7

Suppose that is a tensor rank function. We say that is a strongly proper tensor rank function if for any with , and , we have

(2.2)

We cannot change submax in (2.2) to the third largest value of as this violates Properties 1 and 3 of Definition 2.1.

We will show that is strongly proper in the next section.

3 CP Rank, Max-Tucker Rank and Submax-Tucker Rank

As we stated in the introduction, our motivation to introduce the axiom system for tensor ranks is to find some tensor ranks which have the sub-full-rank property and the base tensor property. The six properties of Definition 2.1 are not satisfied by some tensor ranks in the literature. For example, the tubal rank of third order tensors was introduced in [5]. For , . Thus, it is not a tensor rank function even for third order tensors. It is still very useful in applications [12, 13, 15, 14, 17].

However, the six properties of Definition 2.1 are satisfied by tensor ranks arising from two most important tensor decompositions – the CP decomposition and the Tucker decomposition.

We now study the CP rank [6].

Definition 3.1

Suppose that and . Suppose that there are for and such that

(3.3)

then we say that has a CP decomposition (3.3). The smallest integer such that (3.3) holds is called the CP rank of , and denoted as .

Theorem 3.2

The CP rank is a subadditive tensor rank function. It is not a proper tensor rank function.

Proof We first show that the CP rank is a tensor rank function. Properties 1, 3 and 4 hold clearly from the definition of the CP rank. Before we show Property 2, we can assert that for all since , where with the unique nonzero entry . In the following, we show Property 2 by induction for . We fix here.

For , reduces to the identity matrix and hence Property 2 is true for such a case. Now we assume that . Then we show .

Assume that with . Then

Here, is the all one vector in . This indicates that since . This contradicts the assumption that and hence .

Hence, Property 2 holds.

For Property 5, we have that if when is a permutation of with . Hence we have Property 5.

For property 6, assume that is a subtensor of . For , let and be a subtensor of by a similar way of from . Then we have that and since are rank-one tensors for . This means that . Hence Property 6 is satisfied and the CP rank is a tensor rank function.

Therefore, the CP rank is a tensor rank function.

Suppose that with and . Let

It holds that

Hence, . This shows that it is subadditive. By [6], the CP rank of a tensor given by Kruskal is between and . Thus, the CP rank is not a proper tensor rank function. .

We now study the Tucker rank. In some papers such as [4], the -rank is called the Tucker rank.

Suppose that and . We may unfold to a matrix for . Denote as for . Then the vector is called the -rank of [6].

The -rank is a vector rank. Hence it does not satisfy Definition 2.1. However, if we define

(3.4)

then we have the following proposition.

Theorem 3.3

The function defined by (3.4) is a proper, subadditive tensor rank function. But it is not strongly proper.

Proof We first show that rank function defined by (3.4) is a tensor rank function. To see this, it suffices to show that Property 1-6 are all satisfied.

1) Suppose that for is a zero tensor. Then are zero matrices for . This implies that for . By (3.4), we have . On the other hand, assume that for some with . This means that for , which means that and hence is a zero tensor.

Suppose that , then for all . This can be seen as follows. Assume that there exists such that , then since . From above analysis, if and only if . This contradicts with .

Let . Then From , we have that is rank-one. From and , we have that for all is also rank-one.

Similarly, we have that for any , is rank-one.

Thus, for some and hence is a rank-one tensor.

Conversely, if is a rank-one tensor, then for some nonzero vectors . Then and for all . Thus .

Based on the above analysis, Property 1 is satisfied.

2) Denote Then is a rectangular matrix which can be partitioned to an -dimensional identity matrix and an zero matrix, for , and hence . Thus, .

3) When , then , and for any . Clearly, and hence .

4) Suppose that . For any , and any , and hence . Hence .

5) Suppose that and is any permuted tensor of . Then will be for . So . Hence and the result holds.

6) Suppose that is a subtensor of . Then for all , will be a submatrix of and since is matrix rank. So .

Now we conclude that defined by (3.4) is a tensor rank function.

It is clear that such a tensor rank function is proper from its definition. Furthermore, we have that such rank is also subadditive since matrix rank is subadditive.

In addition, we consider with where is the identity matrix of three dimension. Hence . Hence we conclude that such a tensor rank function is not strongly proper.

.

Thus, we call this tensor rank function the max-Tucker rank in this paper, and denote it as for any .

Note that the max-Tucker rank naturally arises from applications of the Tucker decomposition when people assume that for and fix the value of [2, 11]. Then this means that tensors of max-Tucker ranks not greater than are used. In the following, we introduce a new tensor rank function, which is also associated with the Tucker decomposition, but is different from the max-Tucker rank. We may replace (3.4) by

(3.5)

Then we have the following theorem.

Theorem 3.4

The function defined by (3.5) is a strongly proper tensor rank function. But it is not subadditive.

Proof We first show that function defined by (3.5) is a tensor rank function. It suffices to show that Property 1-6 are all satisfied.

1) Suppose that for is a zero tensor. Then are zero matrices for all . This implies that , which means that . By (3.5), we have . On the other hand, assume that for some with . This means that for some , , and hence , . Therefore, if and only if

Suppose that , then is not zero tensor and hence then for all . Since is defined by (3.5), there exists such that . Without loss of generality, we assume that for . Let . Similar to discussion in proof of Theorem 3.3, is rank-one for all . Thus

for some . Clearly, such is a rank-one tensor.

Conversely, if is a rank-one tensor, then for some nonzero vectors . Then and for all . Thus .

Based on the above analysis, Property 1 is satisfied.

2) Denote Then is a rectangular matrix which can be partitioned to an -dimensional identity matrix and an zero matrix, for , and hence . Thus, .

3) When , then , and for any . Clearly, and when . Hence is the same as the matrix rank of the corresponding matrix.

4) Suppose that . For any , and any , and hence . Hence .

5) Suppose that and is any permuted tensor of . Then will be for . So . Hence and the result holds.

6) Suppose that is a subtensor of . Then for all , will be a submatrix of and since is matrix rank. So .

Now we conclude that defined by (3.5) is a tensor rank function.

The strongly proper property of such a tensor rank function is clear and hence it suffices to show that it is not subadditive.

Let with and

It is assumed that , and , . Then for and . So since and . Therefore, we conclude that such a tensor rank function is not subadditive.

.

Thus, we call this tensor rank function the submax-Tucker rank in this paper, and denote it as for any .

Proposition 3.5

We have for any and for some . Thus, . Furthermore, is strongly proper.

Proof Clearly, for any . To see for some , we consider the following counterexample .

Consider the tensor with its nonzeros entries . By observation, we have that and , which implies that and the result is arrived here.

As and is strongly proper, is also strongly proper.

.

We cannot replace submax in (3.5) by the third largest value in , as this will violate Properties 1 and 3 of Definition 2.1.

4 Maximum Full Rank Subtensors

In this section, we introduce the concept of maximum full rank subtensors, and define the max-full-rank-subtensor property.

We first define the full rank concept for a tensor rank function. Recall that in matrix theory, there is the concept of full row (column) rank matrices.

Definition 4.1

Suppose that is a tensor rank function. Let with , and . If we have

(4.6)

for some index satisfying , then we say that is of full -row rank, or simply say that is of full rank. In particular, zero tensors are regarded as of full rank.

We then define the max-full-rank-subtensor property for a tensor rank function.

Definition 4.2

Suppose that is a tensor rank function. Let . We call a subtensor of a maximum full rank subtensor of under if is of full rank, and is maximum for any such full rank subtensors of . We say that is of the max-full-rank-subtensor property if for any ,