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【转】Simple Simhashing: Clustering in linear time

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AUTHORS
  • Moulton Ryan

ABSTRACT

Suppose you have a huge number of items that you would like to group together by a fuzzy notion of similarity. Suppose the only tool available to you is a key-value store. Suppose you only have the resources to consider each object once. Never fear, simhashing is here!

Well, there goes the neighborhood…

Most clustering algorithms are frustratingly non-local, and what is frustrating at small scale becomes intractable at large scale. Limiting your scope to a neighborhood of items usually requires heuristics that are clustering algorithms in their own right (Yo dawg, I put some clustering in your clustering.) Any algorithm that requires a notion of pairwise similarity at best requires fetching many items from your data store, and at worse requires textstyle{n^2} time and space.
 
Wouldn’t it be nice if you could look at an item once, and determine its cluster immediately without consulting any other data? Wouldn’t it be nice if the clusters were stable between runs, so that the existence of one item would never change the cluster of another? Simhashing does exactly this. There are many approaches tosimhashing, in this document I’m going to talk only about my favorite. It’s simple to implement, mathematically elegant, works on anything with many binary features, and produces high quality results. It’s also simple to analyze, so don’t let the notation scare you off.
 

Comparing Two Sets

Suppose you have two sets, textstyle{A} and textstyle{B}, and you would like to know how similar they are. First you might ask, how big is their intersection?

 

|Acap B|
 
That’s nice, but isn’t comparable across different sizes of sets, so let’s normalize it by the union of the two sizes.
 
frac{|Acap B|}{|Acup B|}
 
This is called the Jaccard Index, and is a common measure of set similarity. It has the nice property of being 0 when the sets are disjoint, and 1 when they are identical.
 

Hashing and Sorting

Suppose you have a uniform pseudo-random hash function textstyle{H} from elements in your set to the range textstyle{[0, 1]}. For simplicity, assume that the output of textstyle{H} is unique for each input. I’ll use textstyle{H(A)} to denote the set of hashes produced by applying textstyle{H} to each element of textstyle{A}, i.e. textstyle{left{H(a_1),;H(a_2),; H(a_3),;dots;,; H(a_n)right} }.

 
Consider textstyle{min(H(A))}. When you insert and delete elements from textstyle{A}, how often does textstyle{min(H(A))}change?
 
If you delete textstyle{x} from textstyle{A} then textstyle{min(H(A))} will only change if textstyle{H(x)=min(H(A))}. Since any element has an equal chance of having the minimum hash value, the probability of this is textstyle{frac{1}{|A|}}.
 
If you insert textstyle{x} into textstyle{A} then textstyle{min(H(A))} will only change if textstyle{H(x) < min(H(A))}. Again, since any element has an equal chance of having the minimum hash value, the probability of this is textstyle{frac{1}{|A|+1}}.
 
For our purposes, this means that textstyle{min(H(A))} is useful as a stable description of textstyle{A}.
 

Probability of a Match

What is the probability that textstyle{min(H(A))=min(H(B))}?
 
If an element produces the minimum hash in both sets on their own, it also produces the minimum hash in their union.
 
textstyle{min(H(A))=min(H(B))} if and only if textstyle{min(H(Acup B))=min(H(A))=min(H(B))}.
 
Let textstyle{x} be the member of textstyle{Acup B} that produces the minimum hash value. The probability that textstyle{A} and textstyle{B} share the minimum hash is equivalent to the probability that textstyle{x} is in both textstyle{A} and textstyle{B}. Since any element of textstyle{Acup B} has an equal chance of having the minimum hash value, this becomes
 
frac{|Acap B|}{|Acup B|}
 
Look familiar? Presto, we now have a simhash.
 

Tuning for Precision

This may be too generous for your purposes, but it is easy to make it more restrictive. One approach is to repeat the whole process with textstyle{n} independent hash functions, and concatenate the results. This makes the probability of a match
 
left(frac{|Acap B|}{|Acup B|}right)^n
 
I prefer an alternate approach. Use only one hash function, but instead of selecting only the minimum value as the simhash, select the least textstyle{n} values. The probability of a match then becomes

frac{|Acap B|choose{n}}{|Acup B|choose{n}} 
 
and if textstyle{n ll |Acap B|},
 
  frac{|Acap B|choose{n}}{|Acup B|choose{n}} approxleft(frac{|Acap B|}{|Acup B|}right)^n 
 
The advantage of this over independent hash functions is that it sets a minimum on the number of members that the two sets must share in order to match. This mitigates the effect of extremely common set members on your clusters. With several independent hash functions, a very common set member that produces low values in a small number of hash functions can cause a huge blowup of the resulting clusters. Selecting n from a single hash function ensures that it can only effect one term. It is for this reason that many simhash implementations unrelated to this one take into account the global frequency of each feature, but this complicates their implementation.
 

Turning Anything Into a Set

This algorithm works on a set, but the things we’d like to cluster usually aren’t sets. Mapping from one to the other is straightforward if each item has many binary features, but can require some experimentation to get good results. If your items are text documents, you can produce a set using a sliding window of n-grams. I’ve found 3-grams to work well on lyrics, but YMMV. Since there’s no order to the members of the set, it’s important to make them long enough to preserve some of the local structure of the thing you’d like to cluster.
 

Pseudocode

int SimHash(Item item, int restrictiveness)
    Set set = SplitItemToSet(item)
    PriorityQueue queue
    for x in set
        queue.Insert(Hash(x))
    simhash = 0
    for x in [0 : restrictiveness]
        simhash ^= queue.PopMin()
    return simhash

 

Further Reading

This specific technique is often referred to as “Min Hashing” in the literature, so that’s a good query to start looking for specific applications to your problem. It is a member of a general class of techniques called “Locality Sensitive Hashing,” often abbreviated as LSH. Google has patented an application of anothersimhashing technique that is generally unrelated to this one. The paper that introduces it is also available, as is Google’s paper on their implementation.

REFERENCES

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