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 ).
In this article, we show several results obtained by combining the use of stable distributions with pseudorandom generators for bounded space. In particular:---We show that, for any p ∈ (0, 2], one can maintain (using only O(log n e2) words of storage) a sketch C(q) of a point q ∈ lnp under dynamic updates of its coordinates. The sketch has the property that, given C(q) and C(s), one can estimate Vq − sVp up to a factor of (1 p e) with large probability. This solves the main open problem of [1999].---We show that the aforementioned sketching approach directly translates into an approximate algorithm that, for a fixed linear mapping A, and given x ∈ ℜn and y ∈ ℜm, estimates VAx − yVp in O(n p m) time, for any p ∈ (0, 2]. This generalizes an earlier algorithm of Wasserman and Blum [1997] which worked for the case p e 2.---We obtain another sketch function C′ which probabilistically embeds ln1 into a normed space lm1. The embedding guarantees that, if we set m e log(1 Δ)O(1 e), then for any pair of points q, s ∈ ln1, the distance between q and s does not increase by more than (1 p e) with constant probability, and it does not decrease by more than (1 − e) with probability 1 − Δ. This is the only known dimensionality reduction theorem for the l1 norm. In fact, stronger theorems of this type (i.e., that guarantee very low probability of expansion as well as of contraction) cannot exist [Brinkman and Charikar 2003].---We give an explicit embedding of ln2 into lnO(log n)1 with distortion (1 p 1 nΘ(1)).