LSH之p-stable分布

1:Cauchy distribution

Probability density function

The Cauchy distribution has the probability density function

         

 

 

where x₀ is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). γ is 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 when x₀ = 0 and γ = 1 is called the standard Cauchy distribution with the probability density function

 

Cumulative distribution function

The cumulative distribution function (cdf) is:

Cumulative distribution function

 

2:p-stable distributions

 

 

根据上面原理，很容易证明标准正态分布是2-stable。

 

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问题：

1：怎么预先计算k值

通过随机从dataset中取小量点，让后按照算法计算一边，通过递增k值，找到一个k值使得计算时间最小。

2：怎么放bucket里面

每个点，都有L个K元向量，其实向量中的每个元素都是同一种性质的，只是用了不同hash函数hash的话。至于具体怎么分布的就要看h1这个函数了。

3：怎么保证精确度

manual手册上有详细说明，其实为什么作者选用标准正态分布，就是因为标准正态分布是2-stable，这样在精确度方面就有了数学的保证