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We present a new approach to approximate nearest-neighbor queries in fixeddimension under a variety of non-Euclidean distances. We are given a set $S$ of$n$ points in $\mathbb{R}^d$, an approximation parameter $\varepsilon > 0$, anda distance function that satisfies certain smoothness and growth-rateassumptions. The objective is to preprocess $S$ into a data structure so thatfor any query point $q$ in $\mathbb{R}^d$, it is possible to efficiently reportany point of $S$ whose distance from $q$ is within a factor of $1+\varepsilon$of the actual closest point.  Prior to this work, the most efficient data structures for approximatenearest-neighbor searching in spaces of constant dimensionality applied only tothe Euclidean metric. This paper overcomes this limitation through a methodcalled convexification. For admissible distance functions, the proposed datastructures answer queries in logarithmic time using $O(n \log (1 / \varepsilon)/ \varepsilon^{d/2})$ space, nearly matching the best known bounds for theEuclidean metric. These results apply to both convex scaling distance functions(including the Mahalanobis distance and weighted Minkowski metrics) and Bregmandivergences (including the Kullback-Leibler divergence and the Itakura-Saitodistance).

Approximate Nearest Neighbor Searching with Non-Euclidean and Weighted Distances