DOWN THE RABBIT HOLE: ROBUST PROXIMITY SEARCH AND DENSITY ESTIMATION IN SUBLINEAR SPACE

For a set of n points in R-d, and parameters k and epsilon, we present a data structure that answers (1 + epsilon, k) approximate nearest neighbor queries in logarithmic time. Surprisingly, the space used by the data structure is (O) over tilde (n/k), where the (O) over tilde(center dot) notation here hides terms that are exponential in d, roughly varying as 1/epsilon(d); as such, the space used is sublinear in the input size if k is sufficiently large. Our approach provides a novel way to summarize geometric data, such that meaningful proximity queries on the data can be carried out using this sketch. Using this, we provide a sublinear space data structure that can estimate the density of a point set under various measures, including (i) sum of distances of k closest points to the query point and (ii) sum of squared distances of k closest points to the query point. Our approach generalizes to other distance-based estimations of densities of similar flavor. We also study the problem of approximating some of these quantities when using sampling. In particular, we show that a sample of size (O) over tilde (n/k) is sufficient, in some restricted cases, to estimate the above quantities. Remarkably, the sample size has only linear dependency on the dimension.