Efficient Approximate Search for Sets of Vectors

We consider a similarity measure between two sets $A$ and $B$ of vectors, that balances the average and maximum cosine distance between pairs of vectors, one from set $A$ and one from set $B$. As a motivation for this measure, we present lineage tracking in a database. To practically realize this measure, we need an approximate search algorithm that given a set of vectors $A$ and sets of vectors $B1,...,Bn$, the algorithm quickly locates the set $B_i$ that maximizes the similarity measure. For the case where all sets are singleton sets, essentially each is a single vector, there are known efficient approximate search algorithms, e.g., approximated versions of tree search algorithms, locality-sensitive hashing (LSH), vector quantization (VQ) and proximity graph algorithms. In this work, we present approximate search algorithms for the general case. The underlying idea in these algorithms is encoding a set of vectors via a "long" single vector. The proposed approximate approach achieves significant performance gains over an optimized, exact search on vector sets.