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The method of stable random projections is popular for efficiently computingthe Lp distances in high dimension (where 0<p<=2), using small space. Becauseit adopts nonadaptive linear projections, this method is naturally suitablewhen the data are collected in a dynamic streaming fashion (i.e., turnstiledata streams). In this paper, we propose to use only the signs of the projecteddata and analyze the probability of collision (i.e., when the two signsdiffer). We derive a bound of the collision probability which is exact when p=2and becomes less sharp when p moves away from 2. Interestingly, when p=1 (i.e.,Cauchy random projections), we show that the probability of collision can beaccurately approximated as functions of the chi-square similarity. For example,when the (un-normalized) data are binary, the maximum approximation error ofthe collision probability is smaller than 0.0192. In text and visionapplications, the chi-square similarity is a popular measure for nonnegativedata when the features are generated from histograms. Our experiments confirmthat the proposed method is promising for large-scale learning applications.

Sign Stable Projections, Sign Cauchy Projections and Chi-Square Kernels