# Multi-Angle Fast Neural Tangent Kernel Classifier
## 1 Introduction  
Kernel methods, such as support vector machines (SVM) and kernel ridge regression, are an important class of machine learning methods. These methods implicitly map low-dimensional, indistinguishable data points to a high-dimensional space and teach linear learners in high-dimensional areas. This implicit mapping is generated by kernel function induction, so the choice of kernel function can often determine the performance of kernel methods. When the sample features contain heterogeneous information, the sample size is large, the distribution of multidimensional data is irregular (e.g., NON-IID), or the data are not flat in the high-dimensional feature space, it is not reasonable to use a single simple kernel for mapping all samples. To better handle these kinds of data (such as building models on the data collected by personal terminal devices), vertical federated learning (VFL) is widely studied as a variant of federated learning (FL) [1, 2]. The emergence of VFL stems from the concern for data privacy. It summarizes the parameters of each sub-model to train the global model (generally through neural networks). These sub-models are trained by vertical partition data with different attributes (i.e., feature dimensions) [3]. Based on similar ideas, there is also split learning [4]. Different from FL, it breaks the network structure. Each device retains only a part of the network structure, and the subnets of all devices form a complete network model. In the training process, different devices only perform forward or backward calculations on the local network and transmit the calculation results to the next device. Multiple devices complete the model training through the intermediate results of the joint network layer until the model converges [5]. \"In network learning\" is a particular FL structure [6], different from the classic FL. It runs distributed processing through the FLtraining, reasoning and testing stages. The information exchanged by each node in various locations and the bandwidth requirements has also been thoroughly studied [7]. In contrast, using multi-core learning instead of single-core methods enhances the interpretability of the decision function, is more capable of handling large samples and variable data, and efficiently achieves better performance than single-core models [8; 9]. Compared with a single kernel function, the multicore model has higher flexibility. The high-dimensional space after mapping multiple kernel functions is a combinatorial space made by combining various feature spaces, and the combinatorial space can combine different feature mapping capabilities of each subspace to map various features in the data by the most appropriate single kernel function, respectively, which more accurately represents the data in the new combinatorial space. Multi-kernel learning methods have different classification methods according to different classification standards. According to the other construction methods and characteristics of multi-kernel functions, multi-kernel learning methods can be roughly divided into three categories: the composite kernel method combines multiple kernel functions with different factors [10]. It combines various kernel functions by directly summing or weighted summing the essential kernel functions. However, this method may lose some characteristic information of the original data; for all input samples, the weights corresponding to different cores are unchanged, and the samples are virtually averaged. Based on this consideration, a non-stationary combination method [11] is proposed, and each input sample is assigned with different weight coefficients. The multi-scale kernel [12] approach is relatively more flexible and can provide a more complete scale selection than the synthetic kernel method, which is more conducive to processing data with uneven distribution [13]. The above two methods are based on the premise of a linear combination of finite kernel functions, but they are not necessarily effective for some large-scale problems [14].