We introduce a new variant of the nearest neighbor search problem, which allows for some coordinates of the dataset to be arbitrarily corrupted or unknown. Formally, given a dataset of @math points @math in high-dimensions, and a parameter @math , the goal is to preprocess the dataset, such that given a query point @math , one can compute quickly a point @math , such that the distance of the query to the point @math is minimized, when ignoring the "optimal" @math coordinates. Note, that the coordinates being ignored are a function of both the query point and the point returned. We present a general reduction from this problem to answering ANN queries, which is similar in spirit to LSH (locality sensitive hashing) [IM98]. Specifically, we give a sampling technique which achieves a bi-criterion approximation for this problem. If the distance to the nearest neighbor after ignoring @math coordinates is @math , the data-structure returns a point that is within a distance of @math after ignoring @math coordinates. We also present other applications and further extensions and refinements of the above result. The new data-structures are simple and (arguably) elegant, and should be practical -- specifically, all bounds are polynomial in all relevant parameters (including the dimension of the space, and the robustness parameter @math ).
We consider the problem of approximate nearest neighbors in high dimensions, when the queries are lines. In this problem, given n points in Rd, we want to construct a data structure to support efficiently the following queries: given a line L, report the point p closest to L. This problem generalizes the more familiar nearest neighbor problem. From a practical perspective, lines, and low-dimensional flats in general, may model data under linear variation, such as physical objects under different lighting. For approximation 1 + e, we achieve a query time of d3n0.5+t, for arbitrary small t > 0, with a space of d2nO(1 e2+1 t2). To the best of our knowledge, this is the first algorithm for this problem with polynomial space and sub-linear query time.  We consider the Approximate Nearest Line Search (NLS) problem. Given a set L of N lines in the high dimensional Euclidean space Rd, the goal is to build a data structure that, given a query point q ∈ Rd, reports a line l ∈ L such that its distance to the query is within (1 + e) factor of the distance of the closest line to the query point q. The problem is a natural generalization of the well-studied Approximate Nearest Neighbor problem for point sets (ANN), and is a natural first step towards understanding how to build efficient nearest-neighbor data structures for objects that are more complex than points. Our main result is a data structure that, for any fixed e > 0, reports the approximate nearest line in time (d + log N + 1 e)O(1) using O(N + d)O(1 e2) space. This is the first high-dimensional data structure for this problem with poly-logarithmic query time and polynomial space. In contrast, the best previous data structure for this problem, due to Magen [16], required quasi-polynomial space. Up to polynomials, the bounds achieved by our data structure match the performance of the best algorithm for the approximate nearest neighbor problem for point sets.