  Fast approximate nearest neighbor (NN) search in large databases is becoming
popular. Several powerful learning-based formulations have been proposed
recently. However, not much attention has been paid to a more fundamental
question: how difficult is (approximate) nearest neighbor search in a given
data set? And which data properties affect the difficulty of nearest neighbor
search and how? This paper introduces the first concrete measure called
Relative Contrast that can be used to evaluate the influence of several crucial
data characteristics such as dimensionality, sparsity, and database size
simultaneously in arbitrary normed metric spaces. Moreover, we present a
theoretical analysis to prove how the difficulty measure (relative contrast)
determines/affects the complexity of Local Sensitive Hashing, a popular
approximate NN search method. Relative contrast also provides an explanation
for a family of heuristic hashing algorithms with good practical performance
based on PCA. Finally, we show that most of the previous works in measuring NN
search meaningfulness/difficulty can be derived as special asymptotic cases for
dense vectors of the proposed measure.
