## 4 Method
### Comparison between Softmax and NV-Softmax  
Here we compare the effects of the back propagation in original softmax loss and normalized virtual softmax loss.  
The image input enters the network and goes through feature extraction network containing many convolutional and pooling layers. It would be troublesome to derive back propagation in such a complex network structure. Therefore, we use a toy model containing only one linear network and one activation function to derive. Considering that convolutional layers is a superposition of linear layer, and operations of pooling and normalization do not generate gradients, this toy model is without loss of generality.  
Suppose the input of the toy model is the fixed \(\mathbf{Z}\), the weight matrix of the linear network is \(\mathbf{U}\), the bias is \(\mathbf{0}\), the output of the linear network is \(\mathbf{net}\), the activation function is \(\mathbf{f}(\cdot)\), the output of the activation function is \(\mathbf{X}\), and the learning rate is \(\alpha\), shown in Figure 11(a). The variables after one back propagation are \(\mathbf{X}^{{}^{\prime}}\), \(\mathbf{U}^{{}^{\prime}}\),\(\mathbf{net}^{{}^{\prime}}\), respectively. We obtain:\[\begin{split}\mathbf{X}^{{}^{\prime}}&=\mathbf{Z}\cdot \mathbf{U}^{{}^{\prime}T}\\ &=\mathbf{Z}\cdot\left(\mathbf{U}----\alpha\frac{\partial\mathbf{L }}{\partial\mathbf{U}^{{}^{\prime}}}\right)^{T}\\ &=\mathbf{Z}\cdot\left(\mathbf{U}----\alpha\frac{\partial\mathbf{ L}}{\partial\mathbf{X}}\frac{\partial\mathbf{X}}{\partial\mathbf{net}^{{}^{ \prime}}}\frac{\partial\mathbf{net}^{{}^{\prime}}}{\partial\mathbf{U}^{{}^{ \prime}}}\right)^{T}\\ &=\mathbf{Z}\cdot\left(\mathbf{U}----\alpha\frac{\partial \mathbf{L}}{\partial\mathbf{X}}\cdot f^{{}^{\prime}}\left(\mathbf{net}^{{}^{ \prime}}\right)\cdot\mathbf{Z}\right)^{T}\end{split} \tag{18}\]  
The activation function in the feature extraction network Resnet in our model is the rectified linear unit (ReLU), and the ReLU function is a linear function with derivative equal to 1 in the activation interval (Figure 12b); thus,  
\[\begin{split}\mathbf{X}^{{}^{\prime}}&=\mathbf{Z} \mathbf{U}^{T}-\alpha\mathbf{Z}\mathbf{Z}^{T}\left(\frac{\partial L}{\partial \mathbf{X}}\right)^{T}\\ &=\mathbf{X}----\beta\left(\frac  
\[\mathbf{X}_{k}^{\prime} =\mathbf{X}_{k}^{*}+\beta\left(\mathbf{W}_{y_{k}}^{*}---\mathbf{W}_{y_ {k}}^{*}\frac{e^{\mathbf{W}_{y_{k}}^{*}\cdot\mathbf{X}_{k}^{*T}}}{\sum_{i=1}^{C }e^{\mathbf{W}_{i}^{*}\cdot\mathbf{X}_{k}^{*T}}+e^{\mathbf{W}_{v\_add}^{*} \cdot\mathbf{X}_{k}^{*T}}}\right)-\beta\mathbf{W}_{v\_add}^{*}\left(\frac{e^{ \mathbf{W}_{v\_add}^{*}\cdot\mathbf{X}_{k}^{*T}}}{\sum_{i=1}^{C}e^{\mathbf{W} _{i}^{*}\cdot\mathbf{X}_{k}^{*T}}+e^{\mathbf{W}_{v\_add}^{*}\cdot\mathbf{X}_{k }^{*T}}}\right)\] \[\triangleq\mathbf{X}_{k}^{*}+\beta\mathbf{V}_{2}---\beta\mathbf{V} _{3} \tag{23}\]  
Figure 12c shows the position of \(\mathbf{X}_{k}\), \(\mathbf{W}_{y_{k}}\), \(\mathbf{V}_{1}\), \(\mathbf{V}_{2}\), and \(\mathbf{V}_{3}\) in the feature space. \(\mathbf{X}_{k}\) is represented by a pentagram, and the gray dashed semicircle is the unit circle. The blue vector and the gray vector indicate the sample's position trained by NV-softmax loss and original softmax loss, respectively. \(\mathbf{W}_{y_{k}}\) is parallel to \(\mathbf{V}_{1}\) and \(\mathbf{V}_{2}\), and \(\mathbf{X}_{k}\) is parallel to \(\mathbf{V}_{3}\). Here for convenience we hypothesize \(\mathbf{X}_{k}=\mathbf{X}_{k}^{*}\), \(\mathbf{W}_{i}=\mathbf{W}_{i}^{*}\) so \(|\mathbf{V}_{2}|>|\mathbf{V}_{1}|\). It is easy to see that comparing with the one trained by original softmax loss, the network trained by normalized virtual softmax loss can bring the sample closer to its class anchor, leading to a more compressed feature space. Since, in our model, every time, the samples enter into the FC layer they will be normalized, so the final location of the sample trained by different softmax loss are on the unit circle.