text: Recently, the method of b-bit minwise hashing has been applied to large-scale linear learning and sublinear time near-neighbor search. The major drawback of minwise hashing is the expensive preprocessing cost, as the method requires applying (e.g.,) k=200 to 500 permutations on the data. The testing time can also be expensive if a new data point (e.g., a new document or image) has not been processed, which might be a significant issue in user-facing applications. We develop a very simple solution based on one permutation hashing. Conceptually, given a massive binary data matrix, we permute the columns only once and divide the permuted columns evenly into k bins; and we simply store, for each data vector, the smallest nonzero location in each bin. The interesting probability analysis (which is validated by experiments) reveals that our one permutation scheme should perform very similarly to the original (k-permutation) minwise hashing. In fact, the one permutation scheme can be even slightly more accurate, due to the "sample-without-replacement" effect. Our experiments with training linear SVM and logistic regression on the webspam dataset demonstrate that this one permutation hashing scheme can achieve the same (or even slightly better) accuracies compared to the original k-permutation scheme. To test the robustness of our method, we also experiment with the small news20 dataset which is very sparse and has merely on average 500 nonzeros in each data vector. Interestingly, our one permutation scheme noticeably outperforms the k-permutation scheme when k is not too small on the news20 dataset. In summary, our method can achieve at least the same accuracy as the original k-permutation scheme, at merely 1 k of the original preprocessing cost.
  @cite_20: There has been considerable interest in random projections, an approximate algorithm for estimating distances between pairs of points in a high-dimensional vector space. Let A in Rn x D be our n points in D dimensions. The method multiplies A by a random matrix R in RD x k, reducing the D dimensions down to just k for speeding up the computation. R typically consists of entries of standard normal N(0,1). It is well known that random projections preserve pairwise distances (in the expectation). Achlioptas proposed sparse random projections by replacing the N(0,1) entries in R with entries in -1,0,1 with probabilities 1 6, 2 3, 1 6, achieving a threefold speedup in processing time.We recommend using R of entries in -1,0,1 with probabilities 1 2√D, 1-1√D, 1 2√D for achieving a significant √D-fold speedup, with little loss in accuracy.
summary: We are also inspired by the work on very sparse random projections'' @cite_20 . The regular random projection method also has the expensive preprocessing cost as it needs @math projections. The work of @cite_20 showed that one can substantially reduce the preprocessing cost by using an extremely sparse projection matrix. The preprocessing cost of very sparse random projections'' can be as small as merely doing one projection. See http: www.stanford.edu group mmds slides2012 s-pli.pdf for the experimental results on clustering classification regression using very sparse random projections @cite_20 .