The method of random projection (RP) is the standard technique in machine learning and many other areas, for dimensionality reduction, approximate near neighbor search, compressed sensing, etc. Basically, RP provides a simple and effective scheme for approximating pairwise inner products and Euclidean distances in massive data. Closely related to RP, the method of random Fourier features (RFF) has also become popular, for approximating the Gaussian kernel. RFF applies a specific nonlinear transformation on the projected data from random projections. In practice, using the (nonlinear) Gaussian kernel often leads to better performance than the linear kernel (inner product), partly due to the tuning parameter $(γ)$ introduced in the Gaussian kernel. Recently, there has been a surge of interest in studying properties of RFF.
After random projections, quantization is an important step for efficient data storage, computation, and transmission. Quantization for RP has also been extensive studied in the literature. In this paper, we focus on developing quantization algorithms for RFF. The task is in a sense challenging due to the tuning parameter $γ$ in the Gaussian kernel. For example, the quantizer and the quantized data might be tied to each specific tuning parameter $γ$. Our contribution begins with an interesting discovery, that the marginal distribution of RFF is actually free of the Gaussian kernel parameter $γ$. This small finding significantly simplifies the design of the Lloyd-Max (LM) quantization scheme for RFF in that there would be only one LM quantizer for RFF (regardless of $γ$). We also develop a variant named LM$^2$-RFF quantizer, which in certain cases is more accurate. Experiments confirm that the proposed quantization schemes perform well.
Random Fourier Features for Operator-Valued Kernels
IBISC；Université d’Évry val d’Essonne（埃松省埃夫里大学）；LTCI；Télécom ParisTech（巴黎电信学院）；Université Paris-Saclay（巴黎萨克雷大学）；Department of Information and Computer Science（信息与计算机科学系）；Aalto University（阿尔托大学）
Department of Computing and Mathematical Sciences, California Institute of Technology（计算与数学科学系，加利福尼亚理工学院）；Department of Electrical Engineering, California Institute of Technology（电气工程系，加利福尼亚理工学院）
Data Dependent Kernel Approximation using Pseudo Random Fourier Features
IRISA, Univ. Bretagne-Sud（布列塔尼南方大学 IRISA）；ETIS UMR 8051, Univ. Paris Seine, Univ. Cergy-Pontoise, ENSEA, CNRS（ETIS UMR 8051，巴黎塞纳大学，塞尔吉-蓬图瓦兹大学，ENSEA，法国国家科学研究中心）
Towards A Unified Analysis of Random Fourier Features
University of Oxford（牛津大学）；King’s College London（伦敦国王学院）
随机傅立叶特征是一种广泛使用的简单有效的核方法扩展技术。然而，现有的理论分析的方法，仍然集中在特定的学习任务，并通常给出悲观的界限，这是不一致的实证结果。我们解决了这些问题，并提供了第一个统一的风险分析学习与随机傅立叶特征使用平方误差和Lipschitz连续损失函数。在我们的边界，计算成本和预期的风险收敛速度之间的权衡是特定于问题的，并表示在正则化参数和\emph{number of effective degrees of freedom}。我们研究了标准的随机傅立叶特征的方法，我们提高了现有的界限，以保证相应的极小极大风险的核岭回归收敛速度所需的功能的数量，以及一个数据相关的修改，采样功能成比例的\emph{ridge leverage scores}，进一步减少了所需的功能数量。由于岭杠杆分数是昂贵的计算，我们设计了一个简单的近似方案，可证明降低了计算成本，而不损失统计效率。
Convergence for adaptive resampling of random Fourier features
Umeå University（于默奥大学）；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；RWTH Aachen University（亚琛工业大学）；Kungl. Tekniska Högskolan（皇家理工学院）
