A novel CNN framework to extract multi-level modular features for the classification of brain networks

Abstract

Brain disease diagnosis based on brain network classification has become a hot topic. Recently, classification methods based on convolutional neural networks (CNNs) have attracted much attention due to their ability to capture the basic topological structure of the brain network. However, they ignore abnormal structures within modules caused by brain disease, which limits the diagnostic accuracy. In this paper, we propose a novel brain network classification framework based on a CNN model capable of extracting modular features from brain networks at the node and whole-network levels. More specifically, we first develop a novel algorithm to obtain the modular structure of each node, which is then fed into a CNN model to extract the node-level modular features. Second, we minimize the harmonic modularity of the extracted node-level features to reveal the modular structure at the whole-brain network level. Finally, we employ a deep neural network to further extract high-level features for the classification of brain disease. The experimental results on a real-world autism spectrum disorder dataset show that our proposed method achieves the best accuracy of 68.55% and outperforms other common methods and demonstrates the discriminant power of the modular features at multiple levels. In addition, feature analysis based on the trained framework reveals the associations between modular structures and brain disease, which provides new insights into the pathological mechanism from the perspective of modular structures.

Appendices

Appendix A: The gradients of the harmonic modularity regularization term

In this section, we try to give the gradients of the regularization term $\frac {\partial \| F - D^{-1} A F \|_{2, 1}}{\partial F}$. To solve it, we give the gradients of matrix norm L_p,q instead, this norm is defined as

$$ \| X \|_{p,q} = \left[ \sum\limits_{j=1}^n \left( \sum\limits_{i=1}^m | X_{ij} |^p \right)^{\frac{q}{p}} \right]^{\frac{1}{q}} $$

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In order to get the gradients, we need a few more definitions as follows

$$ \begin{array}{@{}rcl@{}} A : X &=& \text{tr} \left( A^T X \right) \\ X^{\odot 2} &=& X \odot X \end{array} $$

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where $\text {tr} \left (X \right ) = {\sum }_i X_{ii}$ indicates the trace of a matrix, A ⊙ B is the Hadamard (element-wise) product. let $A = \text {abs} \left (X \right )$ where the absolute function is also element-wise, we can get

$$ \begin{array}{@{}rcl@{}} X \odot X &=& A \odot A \\ X \odot dX &=& A \odot dA \end{array} $$

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Let N = ∥X∥_p,q, we can obtain the following equations

$$ \begin{array}{@{}rcl@{}} N^q &= &\sum\limits_{j=1}^n \left( \sum\limits_{i=1}^m | X_{ij}|^p \right)^{\frac{q}{p}} \\ &=&\begin{bmatrix} \left( {\sum}_{i=1}^m |X_{i1}|^p \right)^{\frac{q}{p}} & \left( {\sum}_{i=1}^m |X_{i2}|^p \right)^{\frac{q}{p}} & {\cdots} & \left( {\sum}_{i=1}^m |X_{in}|^p \right)^{\frac{q}{p}} \end{bmatrix}^T \\ && \quad \cdot \begin{bmatrix} 1 & 1 & {\cdots} & 1 \end{bmatrix} \\ &=& \left( \begin{bmatrix} {\sum}_{i=1}^m |X_{i1}|^p & {\sum}_{i=1}^m |X_{i2}|^p & {\cdots} & {\sum}_{i=1}^m |X_{in}|^p \end{bmatrix} \right)^{\odot \left( \frac{q}{p} \right)} : 1_n \\ &=& \left( \begin{bmatrix} 1 & 1 & {\cdots} & 1 \end{bmatrix} \cdot \begin{bmatrix} |X_{11}|^p & {\cdots} & |X_{1n}|^p \\ {\vdots} & {\ddots} & {\vdots} \\ |X_{m1}|^p & {\cdots} & |X_{mn}|^p \end{bmatrix} \right)^{\odot \left( \frac{q}{p} \right)} : 1_n \\ &=& \left( 1_m A^{\odot p} \right)^{\odot \left( \frac{q}{p} \right)} : 1_n \end{array} $$

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Therefore,

$$ \begin{array}{@{}rcl@{}} q N^{(q-1)} dN &=& \frac{q}{p} \left( 1_m A^{\odot p} \right)^{\odot \left( \frac{q}{p} -1 \right) } \odot 1_m \left( p A^{\odot \left( p-1 \right)} \odot d A \right) : 1_n\\ \Rightarrow N^{(q-1)} d N &=& \left( 1_m A^{\odot p} \right)^{\odot \left( \frac{q}{p} -1 \right) } \odot 1_m \left( A^{\odot \left( p-1 \right)} \odot d A \right) : 1_n \\ &=& \left( 1_m A^{\odot p} \right)^{\odot \left( \frac{q}{p} -1 \right) } \odot 1_n : 1_m \left( A^{\odot \left( p-1 \right)} \odot d A \right) \\ &=& \left( 1_m A^{\odot p} \right)^{\odot \left( \frac{q}{p} -1 \right) } : 1_m \left( A^{\odot \left( p-1 \right)} \odot d A \right) \\ &=& 1_m^T \left( 1_m A^{\odot p} \right)^{\odot \left( \frac{q}{p} -1 \right) } : A^{\odot \left( p-1 \right)} \odot d A \\ &=& \left( 1_m^T 1_m A^{\odot p} \right)^{\odot \left( \frac{q}{p} -1 \right) } : A^{\odot \left( p-2 \right)} \odot A \odot d A \\ &= &\left( 1_m^T 1_m A^{\odot p} \right)^{\odot \left( \frac{q}{p} -1 \right) }\odot A^{\odot \left( p-2 \right)} : X \odot d X \\ \Rightarrow \frac{\partial N}{\partial X} &=& \left( 1_{m \times m} A^{\odot p} \right)^{\odot \left( \frac{q}{p} -1 \right) }\odot A^{\odot \left( p-2 \right)} \odot X \end{array} $$

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Finally, we substitute the p,q to 2, 1 and get

$$ \frac{\partial \| X \|_{2, 1}}{\partial X} = \left( 1_{m \times m} A^{\odot 2} \right) ^{\odot -\frac{1}{2}} \odot X $$

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Appendix B: Feature analysis

In a neural network model, the strength of the connections between neurons in two consecutive layers (weights) can quantitatively announce the contribution of each neuron in previous layer to the activation of neuron in the last layer. Therefore, we can conduct the relative importance values (RIVs) of neurons in l-th layer F^(l) in a backward manner by well trained weights W^(l) and F^(l+ 1).

Here we summarize the calculation procedure of each layer in formulation. For the fully connected layer

$$ F^{(l)} = W^{(l)} F^{(l+1)} \in \mathbb{R}^{M_l \times 1} $$

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where F^(L) = [1,− 1]^T indicate the RIVs in the last layer L for classification.

The RIVs of N2G layer $F^{(l)} \in \mathbb {R}^{N \times M_l}$ can be obtained by

$$ F^{(l)} =|W^{(l)}| |F^{(l+1)}| $$

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In the CNN-MM model, there are two N2G layers, where the second layer takes the node-level features as input, where node-level features consist of topological features and modular features. Therefore, the corresponding RIVs F^(l) can be divided into two parts. Let the i-th row be $F_i^{(l)} = [F_{i,1}^{(l)}, \cdots , F_{i, M_{l_1}}^{(l)}, F_{i, M_{l_1}+1}^{(l)}, {\cdots } , F_{i, M_l}^{(l)}]$, where $M_{l_1}$ corresponds to the dimension of node-level topological features and the $M_l - M_{l_1}$ represents that of node-level modular features. We can obtain the RIV of k-th modular structure by $F_k^L={\sum }_{j=M_{l_1}+1}^{M_l} F_{k,j}^{(l)}$, which can be used to identify the discriminant power of each brain region from the perspective of modular structures.

For the E2N-EW layer

$$ F_r^{(l)} = | W_r^{(l)}| | \left( F_r^{(l+1)} \right)^T | $$

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where $F_r^{(l)}$, $W_r^{(l)}$ and $F_r^{(l+1)}$ indicate the r-th row of $F^{(l)} \in \mathbb {R}^{N \times N \times M_l}$, $W^{(l)} \in \mathbb {R}^{N \times N \times M_l \times M_l}$ and $F^{(l+1)} \in \mathbb {R}^{N \times M_{l+1}}$respectively. Based on the calculation procedure mentioned above, we can obtain the RIV of each edge in either the brain network A or the modular structure A^(k).