  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 $(\gamma)$ 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 $\gamma$ in the Gaussian kernel. For example, the quantizer
and the quantized data might be tied to each specific tuning parameter
$\gamma$. Our contribution begins with an interesting discovery, that the
marginal distribution of RFF is actually free of the Gaussian kernel parameter
$\gamma$. 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 $\gamma$). 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.
