A Lower Bound on the Euclidean Distance for Fast Nearest Neighbor Retrieval in High-dimensional Spaces

Finding the nearest neighbor among a large collection of high dimensional vectors can be a computationally demanding task. In this paper, we pursue fast vector matching by representing vectors in IRn with lower dimensional projections in IR, m ≤ n. The key to creating and using the representative vectors is a lower bound on the Euclidean distance between arbitrary vectors in IRn based on the submultiplicative property of induced matrix norms. For any non-zero projection matrix A ∈ IR, the bound is proportional to the distance between the projected vectors. We study other existing bounds involving orthogonal transforms and piecewise constant approximation maps in light of this formulation. Additionally, we address the question of how to optimize the projection matrix given a dataset in order to make the bound as tight as possible. Experimental results on a speech database show that exact nearest neighbor computation can be accelerated by a factor of 5 using the proposed bound.