Space Exploration via Proximity Search

We investigate what computational tasks can be performed on a point set in $\Re^d$, if we are only given black-box access to it via nearest-neighbor search. This is a reasonable assumption if the underlying point set is either provided implicitly, or it is stored in a data structure that can answer such queries. In particular, we show the following: (A) One can compute an approximate bi-criteria $k$-center clustering of the point set, and more generally compute a greedy permutation of the point set. (B) One can decide if a query point is (approximately) inside the convex-hull of the point set. We also investigate the problem of clustering the given point set, such that meaningful proximity queries can be carried out on the centers of the clusters, instead of the whole point set

Robust Proximity Search for Balls using Sublinear Space

Given a set of n disjoint balls b1, . . ., bn in IRd, we provide a data structure, of near linear size, that can answer (1 \pm \epsilon)-approximate kth-nearest neighbor queries in O(log n + 1/\epsilon^d) time, where k and \epsilon are provided at query time. If k and \epsilon are provided in advance, we provide a data structure to answer such queries, that requires (roughly) O(n/k) space; that is, the data structure has sublinear space requirement if k is sufficiently large

Down the Rabbit Hole: Robust Proximity Search and Density Estimation in Sublinear Space

For a set of $n$ points in $\Re^d$, and parameters $k$ and \eps, we present a data structure that answers (1+\eps,k)-\ANN queries in logarithmic time. Surprisingly, the space used by the data-structure is \Otilde (n /k); that is, 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: \begin{inparaenum}[(i)] \item sum of distances of $k$ closest points to the query point, and \item sum of squared distances of $k$ closest points to the query point. \end{inparaenum} Our approach generalizes to other distance based estimation 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 \Otilde (n /k) is sufficient, in some restricted cases, to estimate the above quantities. Remarkably, the sample size has only linear dependency on the dimension

Approximating Minimization Diagrams and Generalized Proximity Search

We investigate the classes of functions whose minimization diagrams can be approximated efficiently in \Re^d. We present a general framework and a data-structure that can be used to approximate the minimization diagram of such functions. The resulting data-structure has near linear size and can answer queries in logarithmic time. Applications include approximating the Voronoi diagram of (additively or multiplicatively) weighted points. Our technique also works for more general distance functions, such as metrics induced by convex bodies, and the nearest furthest-neighbor distance to a set of point sets. Interestingly, our framework works also for distance functions that do not comply with the triangle inequality. For many of these functions no near-linear size approximation was known before

Proximity search heuristics for wind farm optimal layout

A heuristic framework for turbine layout optimization in a wind farm is proposed that combines ad-hoc heuristics and mixed-integer linear programming. In our framework, large-scale mixed-integer programming models are used to iteratively refine the current best solution according to the recently-proposed proximity search paradigm. Computational results on very large scale instances involving up to 20,000 potential turbine sites prove the practical viability of the overall approach

Behavioral Cloning via Search in Video PreTraining Latent Space

Our aim is to build autonomous agents that can solve tasks in environments like Minecraft. To do so, we used an imitation learning-based approach. We formulate our control problem as a search problem over a dataset of experts' demonstrations, where the agent copies actions from a similar demonstration trajectory of image-action pairs. We perform a proximity search over the BASALT MineRL-dataset in the latent representation of a Video PreTraining model. The agent copies the actions from the expert trajectory as long as the distance between the state representations of the agent and the selected expert trajectory from the dataset do not diverge. Then the proximity search is repeated. Our approach can effectively recover meaningful demonstration trajectories and show human-like behavior of an agent in the Minecraft environment