Kernel based methods have become popular in a wide variety of machine learning tasks. They rely on the computation of kernel functions, which implicitly transform the data in its input space to data in a very high dimensional space. Efficient application of these functions have been subject to study in the last 10 years. The main focus was on improving the scalability of kernel based methods. In this regard, kernel function approximation using explicit feature maps have emerged as a substitute for traditional kernel based methods. Over the years, various advancements from the theoretical perspective have been made to explicit kernel maps, especially to the method of random Fourier features (RFF), which is the main focus of our work. In this work, the major developments in the theory of kernel function approximation are reviewed in a systematic manner and the practical applications are discussed. Furthermore, we identify the shortcomings of the current research, and discuss possible avenues for future work.

中文翻译：

---

核函数逼近的重大进展

基于内核的方法在各种机器学习任务中变得流行。它们依赖于核函数的计算，该函数将输入空间中的数据隐式转换为非常高维空间中的数据。在过去的 10 年中，人们一直在研究这些功能的有效应用。主要重点是提高基于内核的方法的可扩展性。在这方面，使用显式特征图的核函数逼近已经成为传统基于核的方法的替代品。多年来，从理论角度来看，显式核映射已经取得了各种进步，尤其是随机傅立叶特征 (RFF) 方法，这是我们工作的主要重点。在这项工作中，系统地回顾了核函数逼近理论的主要发展并讨论了实际应用。此外，我们确定了当前研究的缺点，并讨论了未来工作的可能途径。