Fast Nearest Neighbor Preserving Embeddings

We show an analog to the Fast Johnson-Lindenstrauss Transform for Nearest Neighbor Preserving Embeddings in $\ell_2$. These are sparse, randomized embeddings that preserve the (approximate) nearest neighbors. The dimensionality of the embedding space is bounded not by the size of the embedded set n, but by its doubling dimension λ. For most large real-world datasets this will mean a considerably lower-dimensional embedding space than possible when preserving all distances. The resulting embeddings can be used with existing approximate nearest neighbor data structures to yield speed improvements.