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 ∈ ℝD, one can generate x = ∑i=1Duiri, and y = ∑i=1Dviri, where ri ∼ N(0, 1) iid. Then one can estimate the cosine similarity ρ from sgn(x) and sgn(y). In this paper, we study a series of estimators for “sign-full” random projections. First we prove E(sgn(x)y) = √2/πρ, which provides an estimator for ρ. Interestingly this estimator can be substantially improved by normalizing y. Then we study estimators based on E (y−1x≥0 + y+1x<0) and its normalized version. We analyze the theoretical limit (using the MLE) and conclude that, among the proposed estimators, no single estimator can achieve (close to) the theoretical optimal asymptotic variance, for the entire range of ρ. On the other hand, the estimators can be combined to achieve the variance close to that of the MLE. In applications such as near neighbor search, duplicate detection, knn-classification, etc, the training data are first transformed via random projections and then only the signs of the projected data points are stored (i.e., the sgn(x)). The original training data are discarded. When a new data point arrives, we apply random projections but we do not necessarily need to quantize the projected data (i.e., the y) to 1-bit. Therefore, sign-full random projections can be practically useful. This gain essentially comes at no additional cost.