Private Approximate Nearest Neighbor Search for Vector Database Querying

We consider the problem of private approximate nearest neighbor (ANN) search. A user seeks the closest vector to a target query
$q$
among
$M$
vectors stored in a system of
$N$
non-colluding databases. The user aims to retrieve the ANN without revealing information about
$q$
to any of the
$N$
databases. We provide an information-theoretic formulation of the problem and propose a scheme based on a tree-structured ANN search mechanism. The proposed scheme uses a coding-theoretic approach to traverse the branch in the tree structure that leads to the approximately closest vector to
$q$
while guaran-teeing perfect information-theoretic privacy. We prove that our approach achieves a communication cost of
$O(N^{2}M^{\frac{1}{N-1})}$
for
$N$
databases. For large
$M$
, this communication cost is lower than competing cryptographic ANN search protocols.