LSH之p-stable分布

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], ">   

 

 

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

 

 

根據上面原理，很容易證明標準常態分佈是2-stable。

 

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問題：

1：怎麼預先計算k值

通過隨機從dataset中取小量點，讓後按照演算法計算一邊，通過遞增k值，找到一個k值使得計算時間最小。

2：怎麼放bucket裡面

每個點，都有L個K元向量，其實向量中的每個元素都是同一種性質的，只是用了不同hash函數hash的話。至於具體怎麼分布的就要看h1這個函數了。

3：怎麼保證精確度

manual手冊上有詳細說明，其實為什麼作者選用標準常態分佈，就是因為標準常態分佈是2-stable，這樣在精確度方面就有了數學的保證