# Hierarchical Prototypes Polynomial Softmax Loss Function for Visual Classification
## 1 Introduction  
Deep convolutional neural networks have enhanced image classification tasks [1; 2; 3; 4]. However, due to the large number of deep neural network parameters, it is challenging to deploy on edge devices or mobile devices with limited resources. In recent years, studies have aimed at designing and studying lightweight neural networks with relatively low parameters and calculations [5; 6; 7]. These lightweight neural networks can be widely deployed on mobile terminals and embedded devices, thereby becoming current research hotspots. Studies on lightweight neural networks focus on network design structures and improvement of the loss function [8; 9]. Various advances in network design structures, such as Mobilenetv1 [5], Mobilenetv2 [10] and Shufflenet [6] have been reported. Based on evolutionary neural networks, the design of the network structure commonly used in automatic search methods is MobilenetV3 [11]. However, due to the large amount of computing resources, this method has not been evaluated by various studies. The loss function can improve the model performance without increasing the number of parameters and calculations in the inference stage of the model, therefore, we evaluated the loss function of lightweight networks. Lightweight convolution neural networks and large-scale deep convolution neural network models are both end-to-end methods. Therefore, generally, lightweight networks directly use the loss function of large-scale convolutional neural networks. Currently, there are three main types of loss functions used for network model training. These are the cross-entropy loss functions based on sample labels and its variants, the contrast loss functions based on sample pairing, and the loss functions based on regularization. Regarding the cross-entropy loss function and its variants of sample labels, the input image is first mapped to a vector space through superposition of a multi-layer convolutionalneural network to obtain a feature vector representation of the image. To determine the category of the input image, we usually measure the similarity between the feature vector of the image and class prototypes (also referred to as class weight vectors, class centers or class representations), and then calculate the class of the sample based on the class prototype distance of the sample. Through appropriate measurement methods such as inner product similarity, cosine similarity [12] and Euclidean distance [13] among others, the logits of the category can be calculated by Softmax to obtain the probability of the category the sample belongs to. In addition to Softmax, studies have also proposed the Bayesian radial basis network, Bayesian formula and Gaussian distribution. In the training process, the class probability of the sample is continuously optimized by cross-entropy loss to approximate its ground-truth. The contrast loss function based on sample pairing cannot directly replace the Softmax loss function because it cannot obtain an efficient sample pair. Subsequently, DeepID2 [14], center loss [1] and margin loss [15] combined metric loss and traditional Softmax loss for supervise model training, and made some progress in face recognition. However, the performance of these methods on other visual classification tasks is limited. In addition, CPL [16] and LGM [9] algorithms add the regular term of the distance within the class to the Softmax function to regularize the feature distribution of the class prototype. These methods are still limited by additional unstable regularization losses. Although Euclidean distance and inner product distance have good geometric interpretability, visual classification tasks require class prototype vectors and feature vectors to have good separability. In actual tasks, due to the sample complexity and diversity, especially for fine-grained image classification tasks, the class prototype feature vectors are poorly separable in a low-dimensional space.