P-stable distribution of lsh

1: Cauchy Distribution

Probability Density Function

The Cauchy distribution has the probability density function

= {1 \ over \ PI} \ left [{\ gamma \ over (X-x_0) ^ 2 + \ gamma ^ 2} \ right], ">

 

 

WhereX0 is the location parameter, specifying the location of the peak of the distribution, andGammaIs the scale parameter which specifies the half-width at half-maximum (hwhm ).GammaIs also equal to half the interquartile range and is sometimes called the probable error. cauchy himself exploited such a density function in 1827, with infinitesimal scale parameter, in defining a Dirac delta function (see there ).

Probability Density Function

The purple curve is the standard Cauchy Distribution

 

 

The special case whenX0 = 0 andGamma= 1 is calledStandard Cauchy DistributionWith the probability density function

 

Cumulative Distribution Function

The cumulative distribution function (CDF) is:

Cumulative Distribution Function

 

2: p-stable distributions

 

 

According to the above principle, it is easy to prove that the standard normal distribution is 2-stable.

 

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Problem:

1: how to calculate the K value in advance

Obtain a small number of points from dataset at random, calculate one side based on the algorithm, and increase the K value to minimize the computing time by finding a K value.

2: how to store the bucket

Each vertex has a K-element vector. In fact, each element in the vector is of the same nature, but it uses different hash functions. The specific distribution depends on the H1 function.

3: how to ensure accuracy

The manual Manual provides a detailed description of why the author chooses standard normal distribution because the standard normal distribution is 2-stable, which guarantees the accuracy of mathematics.