Formation of Similarity-Reflecting Binary Vectors with Random Binary Projections

Abstract

We propose a transformation of real input vectors to output binary vectors by projection using a binary random matrix with elements {0,1} and thresholding. We investigate the rate of convergence of the distribution of vector components before binarization to the Gaussian distribution as well as its relationship to the estimation error of the angle between the input vectors by the binarized output vectors. It is shown that for the choice of projection parameters that provide nearly-Gaussian distribution, the experimental and analytical errors are close.Keywords: binary random projections, convergence to the Gaussian distribution, estimate of the similarity of vectors.

Estimation of Vectors Similarity by Their Randomized Binary Projections

“…Similarly to [26,27], we consider projection of vectors by a random binary matrix R with elements r ij from the set { } 0 1 , . Random variables (r.v.)…”

“…u i converges to the Gaussian one with zero mean value and unit variance. The convergence rate of u i is analyzed in [27] (for the review of the convergence problem for the general case see [39,40]). The binary output vector is formed by the binarizing threshold transformation of the intermediate vector u z ® :…”

“…The shortcomings of supervised dimension reduction methods have caused the development of the approach to transformation of vector data without adaptation, so-called random projection [19][20][21][22][23][24][25][26][27]. In this method, to transform input vectors into output ones, multiplication by a random matrix is carried out; elements of the matrix are randomly generated and then fixed numbers from some distribution.…”

“…The formation of binary output vectors with controlled sparsity, which can be used to estimate the similarity (angle value) of real-valued input vectors, is considered in [25] for a ternary random matrix with elements from { } -+ 1 0 1 , , , and in [27] for a binary matrix. In the present paper, we will analyze the estimates of the angle, Euclidean distance, and scalar product of input vectors based on a series of similarity characteristics of binary output vectors when a random binary matrix is used for the transformation.…”

“…The results of the dependence of the error of angle estimate on the dimension of input and output vectors are presented in[27], where the dependence of the error on the angle between input vectors was analyzed.Totransform input real-valued vectors x and y into intermediate ones u and v, random matrices with binary { were used; the probability of a nonzero element of matrices is q = { } 0 5 0 1 . , .…”