Github user jkbradley commented on the issue:
https://github.com/apache/spark/pull/15800
Limiting this discussion to MinHash:
Say we construct a hash table of L x K functions (for doing both OR and AND
amplification). For approxNearestNeighbors, we need to ask, "What is the best
scalar value we can compute from these L*K values to approximate the distance
between keys?"
* I think this is one point of confusion in previous discussion: I am
talking only about nearest neighbors, which sorts on a scalar estimate of
distance. Things are different when you want to build a data structure with
physical hash buckets.
Assume that our LxK functions are chosen independently at random (which is
what we are doing now).
Then I claim that, to compare distances between 2 points (1 query + 1 in
the dataset), there is no better way to combine these functions than to:
* Compute LxK binary 0/1 indicators of whether the 2 points match on each
hash function
* Average these LxK 0/1 indicators
Rough argument:
* For a given scalar hash function (without amplification), there is no
concept of "distance" between hash buckets. (This is different from P-Stable
LSH.)
* The distance metric we want to sort by is Jaccard similarity.
* The average of these indicators is an efficient estimator (in the
statistical sense) of the Jaccard similarity.
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