Probability that a candidate pair will be detected with LSH

Functions to help choose the correct parameters for the lsh and minhash_generator functions. Use lsh_threshold to determine the minimum Jaccard similarity for two documents for them to likely be considered a match. Use lsh_probability to determine the probability that a pair of documents with a known Jaccard similarity will be detected.

Usage

lsh_probability(h, b, s)

lsh_threshold(h, b)

Arguments

h

The number of minhash signatures.

b

The number of LSH bands.

s

The Jaccard similarity.

Details

Locality sensitive hashing returns a list of possible matches for similar documents. How likely is it that a pair of documents will be detected as a possible match? If h is the number of minhash signatures, b is the number of bands in the LSH function (implying then that the number of rows r = h / b), and s is the actual Jaccard similarity of the two documents, then the probability p that the two documents will be marked as a candidate pair is given by this equation.

$$p = 1 - (1 - s^{r})^{b}$$

According to MMDS, that equation approximates an S-curve. This implies that there is a threshold (t) for s approximated by this equation.

$$t = \frac{1}{b}^{\frac{1}{r}}$$

References

Jure Leskovec, Anand Rajaraman, and Jeff Ullman, Mining of Massive Datasets (Cambridge University Press, 2011), ch. 3.

Examples

# Threshold for default values
lsh_threshold(h = 200, b = 40)
#> [1] 0.4781762

# Probability for varying values of s
lsh_probability(h = 200, b = 40, s = .25)
#> [1] 0.03832775
lsh_probability(h = 200, b = 40, s = .50)
#> [1] 0.7191538
lsh_probability(h = 200, b = 40, s = .75)
#> [1] 0.9999803