Tucker Decomposition Network: Expressive Power and Comparison

05/23/2019 ∙ by Ye Liu, et al. ∙ Hong Kong Baptist University 0

Deep neural networks have achieved a great success in solving many machine learning and computer vision problems. The main contribution of this paper is to develop a deep network based on Tucker tensor decomposition, and analyze its expressive power. It is shown that the expressiveness of Tucker network is more powerful than that of shallow network. In general, it is required to use an exponential number of nodes in a shallow network in order to represent a Tucker network. Experimental results are also given to compare the performance of the proposed Tucker network with hierarchical tensor network and shallow network, and demonstrate the usefulness of Tucker network in image classification problems.

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1 Introduction

Deep neural networks have achieved a great success in solving many practical problems. Deep learning methods are based on multiple levels of representation in learning. Each level involves simple but non-linear units for learning. Many deep learning networks have been developed and applied in various applications successfully. For example, convolutional neural networks (CNNs)

[17, 21, 29]

have been well applied in computer vision problems, recurrent neural networks (RNNs)

[9, 12, 25]

are used in audio and natural language processing. For more detailed discussions, see

[22] and its references.

In the recent years, more and more works focus on the theoretical explanations of neural networks. One important topic is the expressive power, i.e., comparing the expressive ability of different neural networks architectures. In the literature [7, 8, 15, 16, 24, 26, 27, 28, 30], researches have been done in the investigation of the depth efficiency of neural networks. It is natural to claim that a deep network can be more powerful in the expressiveness than a shallow network. Recently, Khrulkov et al. [19] applied a tensor train decomposition to exploit the expressive power of RNNs experimentally. In [4], Cohen et al. theoretically analyzed specific shallow convolutional network by using CP decomposition and specific deep convolutional network based on hierarchical tensor decomposition. The result of the paper is that the expressive power of such deep convolutional networks is significantly better than that of shallow networks. Cohen et al. in [5]

generalized convolutional arithmetic circuits into convolutional rectifier networks to handle activation functions, like ReLU. They showed that the depth efficiency of convolutional rectifier networks is weaker than that of convolutional arithmetic circuits.

Although many attempts in theoretical analysis success, the understanding of expressiveness is still needed to be developed. The main contribution of this paper is that a new deep network based on Tucker tensor decomposition is proposed. We analyze the expressive power of the new network, and show that it is required to use an exponential number of nodes in a shallow network to represent a Tucker network. Moreover, we compare the performance of the proposed Tucker network, hierarchical tensor network and shallow network on two datasets (Mnist and CIFAR) and demonstrate that the proposed Tucker network outperforms the other two networks.

The rest of this paper is organized as follows. In Section 2, we briefly review tensor decompositions. We present the proposed Tucker network and show its expressive power in Section 3. In Section 4, experimental results are presented to demonstrate the performance of the Tucker network. Some concluding remarks are given in Section 5.

2 Tensor Decomposition

A -dimensional tensor is a multidimensional array, i.e., . Its -th unfolding matrix is defined as . Given an index subset and the corresponding compliment set , , the -matricization of is denoted as a matrix , obtained by reshaping tensor into matrix.

We also introduce two important operators in tensor analysis, tensor product and Kronecker product. Given tensors and of order and respectively, the tensor product is defined as . Note that when

, the tensor product is the outer product of vectors.

denotes Kronecker product which is an operation on two matrices, i.e., for matrices , , , defined by .

Moreover, we use to denote the set for simplicity.

In the following, we review some well-known tensor decomposition methods and related convolutional networks.

CP decomposition: [3, 14] Given a tensor , the CANDECOMP/PARAFAC decomposition (CP) is defined as follows:

(1)

where , . The minimal value of such that CP decomposition exists is called the CP rank of denoted as .

Tucker decomposition:[6, 31] Given a tensor , the Tucker decomposition is defined as follows:

which can be written as,

(2)

where , , , , , , . The minimal value of such that (2) holds is called Tucker rank of , denoted as . If , we simplicity denoted as .

HT decomposition: The Hierarchical Tucker (HT) Tensor format is a multilevel variant of a tensor decomposition format. The definition requires the introduction of a tree. For detailed discussion, see [10, 11, 13]. Given a tensor , . The hierarchical tensor decomposition has the following form:

(3)

where are the generated vectors of tensor . refer to level- rank. We denote . If all the ranks are equal to , for simple.

2.1 Convolutional Networks

Figure 1: A CP network.
Figure 2: A example of HT network with .

Given a dataset of pairs , each object is represented as a set of vectors with . By applying parameter dependent functions , we construct a representation map . Object with

is classified into one of categories

. Classification is carried out through the maximization of the following score function:

(4)

where is a trainable coefficient tensor.

The representation functions

have many choices. For example, neurons-type functions

for parameters and point-wise non-linear activation . We list some commonly used activation functions here, for example hard threshold: for

, otherwise 0; the rectified linear unit (ReLU)

; and sigmoid .

The main task is to estimate the parameters

and the coefficient tensors . The computational challenge is that the coefficient tensor has an exponential number of entries. We can utilize tensor decompositions to address this issue.

If the coefficient tensor is in CP decomposition, the network corresponding to CP decomposition is called shallow network(or CP Network), see Figure 1. We obtain its score function:

(5)

Note that the same vectors are shared across all classes . If set , the model is universal, i.e., any tensors can be represented.

If the coefficient tensors are in HT format like (8), the network refer to HT network. An example of HT network with is showed in Figure 2. Cohen et al. [4] analyzed the expressive power of HT network and proved that a shallow network with exponentially large width is required to emulate a HT network.

3 Tucker Network

In this section, we propose a Tucker network. If the coefficient tensors in (4) are in Tucker format (2), we refer it as Tucker network, i.e.,

(6)

Suppose for same vectors () in (6). Here be the -th unfolding of tensor . If set , the number of parameter is: . If set , the model is universe, any tensor can be represented by Tucker format, number of parameters are needed. Note that the score function for Tucker network:

The Tucker network architecture is given in Figure 3. The outputs from convolution layer are

where , . The last output, i.e., score value is given as follows:

where is tensor scalar product, i.e., the sum of entry-wise product of two tensors. Because is a order tensor of smaller dimension , it can be further decomposed with a deeper network. In this sense, Tucker network is also a kind of deep network.

Figure 3: Tucker network

The following theorem demonstrates the expressive power of Tucker network.

Theorem 1.

Let be a tensor of order and dimension in each mode, generated by Tucker form in (6). Define for all possible subsets , consider the space of all possible configurations for parameters. In the space, will have CP rank of at least almost everywhere, i.e.,the Lebesgue measure of the space whose CP rank is less than is zero.

The proof can be found in the supplementary section. We remark that if , when is even, the Lebesgue measure of the Tucker format space whose CP rank is less than is zero; when

is odd, the Lebesgue measure of the Tucker format space whose CP rank is less than

is also zero.

3.1 Connection with HT Network

In this subsection, to compare the expressive power of HT and Tucker network, we discuss the relationship between Tucker format and hierarchical Tucker tensor format firstly. Here we only consider hierarchical tensor format, its corresponding HT network (8) has been well discussed in [4].

We start it from hierarchical Tucker tensor, its HT network architecture is shown in Figure 2 . Given a order tensor, its hierarchical tensor format can always be written as

are vectors size of . Here we suppose that . Denote , we have , where

is a linear transformation that converts a matrix into a column vector.

is diagonal operator that transform a vector into a diagonal matrix. Similarly, we have,

where , , and , .

From the property of Kronecker product: , we deduce that,

Therefore,

with , We can get that

(7)

Therefore,

with . It implies that a hierarchical tensor format can be written as a order Tucker tensor. Worth to say, from (7), the rank of is less than that of its factor matrices. Because of the structure of , , we get that and also . From the rank property, .

When the hierarchical tensor has layers, we can similarly deduced the following results.

Theorem 2.

Any hierarchical tensor can be represented as a order Tucker tensor and vice versa.

Theorem 3.

For any tensor , if , then .

The detailed proofs of Theorem 2 and Theorem 3 can be found in Appendix.

According to Theorem 3, given a hierarchical Tucker network of width , we know that the width of Tucker network is not possible larger than .

4 Experimental Results

We designed experiments to compare the performance of three networks: Tucker network, HT network and shallow network. The results illustrate the usefulness of Tucker network. We implement shallow network, Tucker network and HT network with TensorFlow[1] back-end, and test three networks on two different data sets: Mnist [23] and CIFAR-10 [20]. All three networks are trained by using the back-propagation algorithm. In all three networks, we choose ReLU as the activation function in the representation layer

and apply batch normalization

[18] between convolution layer and pooling layer to eliminate numerical overflow and underflow.

We choose Neurons-type with ReLU nonlinear activation as representation map : . Actually the representation mapping now is acted as a convolution layer in general CNNs. Each image patch is transformed through a representation function with parameter sharing across all the image patches. Convolution layer in Figure 3 actually can been seen as a locally connected layer in CNN. It is a specific convolution layer without parameter sharing, which means that the parameters of filter would differ when sliding across different spatial positions. In the hidden layer, without overlapping, a 3D convolution operator size of is applied. Following is a product pooling layer to realize the outer product computation . It can be explained as a pooling layer with local connectivity property, which only connects a neuron with partial neurons in the previous layer. The output of neuron is the multiplication of entries in the neurons connected to it. The fully-connected layer simply apply the linear mapping on the output of pooling layer. The output of Tucker network would be a vector corresponding to class scores.

4.1 Mnist

The MNIST database of handwritten digits has a training set of 60000 examples, and a test set of 10000 examples with 10 categories from 0 to 9. Each image is of

pixels. In the experiment, we select the gradient descent optimizer for back-propagation with batch size 200, and use a exponential decay learning rate with 0.2 initial learning rate, 6000 decay step and 0.1 decay rate. Figure 4

shows the training and test accuracy of three networks with 3834 number of parameters that have been learned. The parameters contains four parameters in batch normalization (mean, std, alpha, beta). We list filter size, strides size and rank as well in Table

1. It is obvious that Tucker network outperforms shallow network and HT network. Moreover, we test the sensitivity of Tucker network with the change of rank, and compare the performance with the other two networks with the same number of parameters. Figure 5 illustrates the sensitivity performance, each value records the highest accuracy in training or test data. Tucker network can achieve the highest accuracy at most times.

4.2 Cifar-10

CIFAR-10 data [20] is a more complicated data set consisting of 60000 color images size of with 10 classes. Here, we use the gradient descent optimizer with 0.05 learning rate and 200 batch size to train. In Figure 6 we report the training and test accuracy with 23790 trained parameters. Table 2 shows the parameter details of sensitivity test, whose results are displayed in Figure 7 . From Figure 6 and Figure 7 , Tucker network still has more excellent performance when fitting a more complicated data set.

5 Conclusion

In this paper, we presented a Tucker network and prove the expressive power theorem. We stated that a shallow network of exponentially large width is required to mimic Tucker network. A connection between Tucker network and HT network is discussed. The experiments on Mnist and CIFAR-10 data show the usefulness of our proposed Tucker network.

Figure 4: Training(left) and testing(right) accuracy of Tucker network for Mnist data.
Figure 5: The performance of Tucker network, Shallow network and Hierarchical tensor network with the change of rank in Mnist data: training accuracy(left); testing accuracy(right).
Network
Num of
parameters
Num of representation
function
Filter size Strides size
Rank
Tucker 10 14 23 14 5 2
HT 3478 14 14 14 14 14 8
Shallow 10 16 21 12 7 2
Tucker 12 14 17 14 11 3
HT 3834 18 14 14 14 14 3
Shallow 16 14 16 14 12 3
Tucker 12 14 15 14 13 4
HT 5300 12 16 26 12 2 4
Shallow 10 20 21 8 7 4
Tucker 11 14 14 14 14 5
HT 8657 11 26 27 2 1 11
Shallow 17 20 23 8 5 10
Table 1: The parameters setting of Fig. 5 in Tucker network, Shallow network and Hierarchical tensor network.
Figure 6: Training(left) and testing(right) accuracy of CIFAR-10 data.
Figure 7: The performance of Tucker network, Shallow network and HT network with the change of rank in CIFAR-10 data: training accuracy(left); testing accuracy(right).
Network
Num of
parameters
Num of
representation  function
Filter size Strides size
Rank
Tucker 10 16 26 16 6 3
HT 13432 10 2121 11 11 3
Shallow 10 17 26 15 6 3
Tucker 10 16 31 16 1 4
HT 17626 22 16 16 1616 6
Shallow 10 20 29 123 4
Tucker 20 16 18 1614 5
HT 23970 30 16 16 1616 6
Shallow 12 25 26 76 19
Tucker 24 16 16 1616 6
HT 32016 12 28 31 41 9
Shallow 20 17 31 151 4
Tucker 31 16 17 1615 7
HT 50233 43 18 21 1411 7
Shallow 37 17 26 156 7
Table 2: The parameters setting of Fig. 7 in Tucker network, Shallow network and Hierarchical tensor network.

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Appendix A Appendix A. Proofs

a.1 Proof of Theorem 1

In section 3, we presented Tucker network and showed its expressive power. To prove Theorem 1, we firstly state and prove three lemmas which will be needed for the proofs.

Lemma 1.

For any matricization of a tensor whose CP rank is ,

Proof.

In Lemma 1, we give the lower bound of the CP-rank. If the matricization of a tensor has matrix rank , then using the above lemma, we get that the CP rank of is larger than .

For a order tensor who is in Tucker format, its matricization has the following form.

Lemma 2.

Given a order tensor whose Tucker format is , index subsets and , then

Proof.

Therefore,

where . Then,

For simplicity, we denote

where

We get

Lemma 3.

If each factor matrix of tensor has full column rank, i.e., has full column rank, then .

Proof.

. ∎

Proof of Theorem 1

Proof.

According to Lemma 1, it suffices to prove that the rank of is at least almost everywhere. From Lemma 3, equivalently, we prove the rank of is at least almost everywhere.

For any , and all possible subsets and the corresponding compliment set , . We let , which simply holds the elements of . Because for all possible subsets . For all , we have . In the following, we will prove that the Lebesgue measure of the space that is zero.

Let be the top-left sub matrix of and is the determinant, as we know that is a polynomial in the entries of , according to theorem in[2], it either vanishes on a set of zero measure or it is the zero polynomial. It implies that the Lebesgue measure of the space whose is zero, i.e., the Lebesgue measure of the space whose rank less than is zero. The result thus follows. ∎

a.2 Proof of Theorem 2

In this subsection, we will prove Theorem 2, the connection of Tucker tensor format and hierarchical Tucker tensor format. The expressive power of hierarchical Tucker tensor network has been well discussed in [4].

In Section 2, we defined -matricization which is a kind of general matricization. In the following, we simply consider the proper order matricization of tensor, denoted as the matrix here, for example, for , ; for , .

The hierarchical tensor decomposition format is given as follows:

(8)