SignRFF: Sign Random Fourier Features

The industry practice has been moving to embedding based retrieval (EBR). For example, in many applications, the embedding vectors are trained by some form of two-tower models. During serving phase, candidates (embedding vectors) are retrieved according to the rankings of cosine similarities either exhaustively or by approximate near neighbor (ANN) search algorithms. For those applications, it is natural to apply ``sign random projections'' (SignRP) or variants, on the trained embedding vectors to facilitate efficient data storage and cosine distance computations. SignRP is also one of the standard indexing schemes for conducting approximate near neighbor search. In the literature, SignRP has been popular and, to an extent, becomes the default method for ``locality sensitive hashing'' (LSH). In this paper, we propose ``sign random Fourier features'' (SignRFF) as an alternative to SignRP. The original method of random Fourier features (RFF) is a standard technique for approximating the Gaussian kernel (as opposed to the linear cosine kernel), in the literature of large-scale machine learning. Basically, RFF applies a simple nonlinear transformation on the samples generated by random projections (RP). Thus, in the pipeline of EBR, it is straightforward to replace SignRP by SignRFF. This paper explains, in a principled manner, why it makes sense to do so. In this paper, a new analytical measure called \textbf{Ranking Efficiency (RE)} is developed, which in retrospect is closely related to the ``two-sample mean'' $t$-test statistic for binomial variables. RE provides a systematic and unified framework for comparing different LSH methods. We compare our proposed SignRP with SignRP, KLSH (kernel LSH), as well SQ-RFF (which is another 1-bit coding scheme for RFF). According to the RE expression, SignRFF consistently outperforms KLSH (for Gaussian kernel) and SQ-RFF. SignRFF also outperforms SignRP in the relatively high similarity region. The theoretical comparison results are consistent with our empirical findings. In addition, experiments are conducted to compare SignRFF with a wide range of data-dependent and deep learning based hashing methods and show the advantage of SignRFF with a sufficient number of hash bits.