Locality Sensitive Hashing Based Clustering

The basic idea of the LSH (Gionis et al. 1999) technique is using multiple hash functions to hash the data points and guarantee that there is a high probability of collision for points which are close to each other and low collision probability for dissimilar points. LSH schemes exist for many distance measures, such as Hamming norm, L _p norms, cosine distance, earth movers distance (EMD), and Jaccard coefficient.

In LSH, define a family H = {h : S → U} as locality-sensitive, if for any a, the function $p(t) = Pr_{H}[h(a) = h(b) :\vert \vert a - b\vert \vert = x]$ is decreasing in x. Based on this definition, the probability of collision of points a and b is decreasing with their distance.

Although LSH was originally proposed for approximate nearest neighbor search in high dimensions, it can be used for clustering as well (Das et al. 2007; Haveliwala et al. 2000). The buckets could be used as the bases for clustering. Seeding the hash functions several times can help getting better...