High-dimensional approximate r-nets

Abstract

The construction of r-nets offers a powerful tool in computational and metric geometry. We focus on highdimensional spaces and present a new randomized algorithm which efficiently computes approximate rnets with respect to Euclidean distance. For any fixed > 0, the approximation factor is 1 + and the complexity is polynomial in the dimension and subquadratic in the number of points. The algorithm succeeds with high probability. Specifically, we improve upon the best previously known (LSHbased) construction of Eppstein et al. [EHS15] in terms of complexity, by reducing the dependence on , provided that is sufficiently small. Our method does not require LSH but, instead, follows Valiant's [Val15] approach in designing a sequence of reductions of our problem to other problems in different spaces, under Euclidean distance or inner product, for which r-nets are computed efficiently and the error can be controlled. Our result immediately implies efficient solutions to a number of geometric problems in high dimension, such as finding the (1 + )-approximate kth nearest neighbor distance in time subquadratic in the size of the input.

High Dimensional Clustering with r-nets

“…This paper presents new theoretical results on the construction of (1 + ǫ)-approximate r-nets, improving the previous upper bound ofÕ(dn 2−Θ( √ ǫ) ) by (Avarikioti et al 2017). We denote n as the number of data points, d the dimension of the data and α = Ω(ǫ 1 3 / log( 1 ǫ )) for an arbitrary error parameter ǫ.…”

“…A major drawback of the framework is its restriction to a constant number of features. Consequentially, this framework was later extended by (Avarikioti et al 2017) to also solve higher dimensional cases. The algorithm, constructed in this paper, yields an immediate improvement on this framework, as the construction of the framework is based around approximate r-nets.…”

“…We also modify our algorithm to yield an improvement on the Net & Prune framework of (Avarikioti et al 2017). Supplying the framework with certain data structures, which are created using approximate r-nets, we derive new algorithms with improved runtime on (1 + ǫ)-approximate k-smallest nearest neighbor distance, (4 + ǫ)-approximate Min-Max Clustering, introduced in (Har-Peled and Raichel 2015), and (4 + ǫ)-approximate k-center clustering.…”

“…Intuitively, we randomly pick a center point from our dataset and then remove every point that is closer then r + ǫd from the center, by checking every point of the dataset. This technique was originally introduced in (Avarikioti et al 2017). The proof of Theorem 1 closely follows this work.…”

“…Building a Net Now, we present how we can build an approximate r-net for a data set, as in (Avarikioti et al 2017): we first employ the sparsification technique and then build the distance matrix in the sparse dataset where we can search efficiently. The running time of building an approximate rnet is dominated by the time complexity of the construction of the distance matrix.…”