Sign-Full Random Projections

The method of 1-bit ("sign-sign") random projections has been a popular tool for efficient search and machine learning on large datasets. Given two $D$-dim data vectors $u$, $v\in\mathbb{R}^D$, one can generate $x = \sum_{i=1}^D u_i r_i$, and $y = \sum_{i=1}^D v_i r_i$, where $r_i\sim N(0,1)$ iid. The "collision probability" is ${Pr}\left(sgn(x)=sgn(y)\right) = 1-\frac{\cos^{-1}\rho}{\pi}$, where $\rho = \rho(u,v)$ is the cosine similarity. We develop "sign-full" random projections by estimating $\rho$ from (e.g.,) the expectation $E(sgn(x)y)=\sqrt{\frac{2}{\pi}} \rho$, which can be further substantially improved by normalizing $y$. For nonnegative data, we recommend an interesting estimator based on $E\left(y_- 1_{x\geq 0} + y_+ 1_{x<0}\right)$ and its normalized version. The recommended estimator almost matches the accuracy of the (computationally expensive) maximum likelihood estimator. At high similarity ($\rho\rightarrow1$), the asymptotic variance of recommended estimator is only $\frac{4}{3\pi} \approx 0.4$ of the estimator for sign-sign projections. At small $k$ and high similarity, the improvement would be even much more substantial.