In this work, we address both of the above issues.

We present an intriguing discovery related to Random Fourier Features: in Gaussian kernel approximation, replacing the random Gaussian matrix by a properly scaled random orthogonal matrix significantly decreases kernel approximation error. We call this technique Orthogonal Random Features (ORF), and provide theoretical and empirical justification for this behavior. Motivated by this discovery, we further propose Structured Orthogonal Random Features (SORF), which uses a class of structured discrete orthogonal matrices to speed up the computation.

Random features have been introduced to scale up kernel methods via randomization techniques. In particular, random Fourier features and orthogonal random features were used to approximate the popular Gaussian kernel. The former is performed by a random Gaussian matrix and leads exactly to the Gaussian kernel after averaging. In this work, we analyze the bias and the variance of the kernel approximation based on orthogonal random features which makes use of Haar orthogonal matrices. We provide explicit expressions for these quantities using normalized Bessel functions and derive sharp exponential bounds supporting the view that orthogonal random features are more informative than random Fourier features.

We present an intriguing discovery related to Random Fourier Features: replacing multiplication by a random Gaussian matrix with multiplication by a properly scaled random orthogonal matrix significantly decreases kernel approximation error. We call this technique Orthogonal Random Features (ORF), and provide theoretical and empirical justification for its effectiveness. Motivated by the discovery, we further propose Structured Orthogonal Random Features (SORF), which uses a class of structured discrete orthogonal matrices to speed up the computation. The method reduces the time cost from $\mathcal{O}(d^2)$ to $\mathcal{O}(d \log d)$, where $d$ is the data dimensionality, with almost no compromise in kernel approximation quality compared to ORF. Experiments on several datasets verify the effectiveness of ORF and SORF over the existing methods. We also provide discussions on using the same type of discrete orthogonal structure for a broader range of kernels and applications.

We present an intriguing discovery related to Random Fourier Features: in Gaussian kernel approximation, replacing the random Gaussian matrix by a properly scaled random orthogonal matrix significantly decreases kernel approximation error. We call this technique Orthogonal Random Features (ORF), and provide theoretical and empirical justification for this behavior. Motivated by this discovery, we further propose Structured Orthogonal Random Features (SORF), which uses a class of structured discrete orthogonal matrices to speed up the computation. The method reduces the time cost from $\mathcal{O}(d^2)$ to $\mathcal{O}(d \log d)$, where $d$ is the data dimensionality, with almost no compromise in kernel approximation quality compared to ORF. Experiments on several datasets verify the effectiveness of ORF and SORF over the existing methods. We also provide discussions on using the same type of discrete orthogonal structure for a broader range of applications.