# Quotient Network-A Network Similar to ResNet but Learning Quotients
## Appendix A  
In this section, we verify the design guidelines presented in Section 3.3 of the main text by conducting experiments on the CIFAR10 dataset. Specifically, we compare the accuracy of using activation functions with different value ranges, whether the activation functions have negative values, whether the activation functions pass through the \((0,1)\) point, whether the activation functions are globally differentiable, and whether the activation functions are used at the beginning of the network, as well as in the shortcuts when increasing the number of channels. The value range of the activation functionWe use 20-layer quotient networks as the basis for comparison. We use two classes of activation functions: the modified linear functions and the modified sigmoid functions. In order to keep the experimental results more comparable, we fix each activation function to pass through the (0, 1) point, the convolutional layer at the beginning of the network, and the shortcuts when increasing the number of channels all use this activation function. The experimental results are shownin Table A1. As the table shows, for the 20-layer networks, the accuracy is the highest when the value range of the modified linear function is [0, 4] or when the value range of the modified sigmoid function is (0, 2). The accuracy will be reduced whether the value range is enlarged or reduced. Especially when the value range is infinite, training cannot be successful. **Whether to contain negative region. ** We continue to use 20-layer networks as the basis for comparison. For the modified linear functions, keep the value range size unchanged at 4; for the modified sigmoid, keep the value range size unchanged at 2. Keep each function passing through the (0, 1) point. The convolution at the beginning of the network and the shortcuts when increasing the number of channels all use this activation function. We only change whether the value range includes the negative area and the size of the negative area, as shown in Table A2. Whether a modified linear function or a modified sigmoid, its accuracy will be reduced when its value range contains the negative area. It can be seen that the larger the area containing negative numbers is, the greater the accuracy decreases. **Whether to pass the point (0, 1). ** Like ResNet, whether it passes the (0, 1) point is the key to whether the deep network can more easily maintain features. So, we use two different depths of 20 and 32 layers for comparison. Similarly, the modified linear function value range is kept as [0, 4], the modified sigmoid function is kept as (0, 2), and the convolution at the beginning of the network and the shortcuts when increasing the number of channels all use the designed activation function; only the function value at 0 is transformed. The comparison of 20-layer networks is shown in Table A3, and the comparison of 32-layer networks is shown in Table A4. When the independent variable is 0, whether the value is greater or less than 1, the accuracy will be reduced, and, as the depth increases, the accuracy decrease will be more obvious. Here, we find that when the modified linear function is used in the 32-layer networks, regardless of whether it passes through (0, 1), the accuracy will be reduced compared with the 20-layer networks. That may be because [0, 4] is no longer suitable for 32-layer networks using the modified linear functions, but it is not important for our experimental purposes. **Globally differentiable or not. ** With the above results, it is easy to realize that the accuracy of the modified linear function is always much lower than the modified sigmoid function. For the quotient network, the modified sigmoid function, which is a globally differentiable function, is more appropriate as the activation function. **The head and shortcuts (channels increasing). ** We finally compare the accuracy changes caused by whether the designed activation function is used in the first convolution and shortcuts when the number of channels increases.