Space Exploration via Proximity Search

Har-Peled, Sariel; Kumar, Nirman; Mount, David M.; Raichel, Benjamin

doi:10.1007/s00454-016-9801-7

Cited by 5 publications

(3 citation statements)

References 21 publications

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“…The following is well known and is included for the sake of completeness, see Reference [13]. It also follows readily from the Preceptron algorithm (see Remark 2.8 below).…”

Section: Algorithm For Approximately Computing the Distance To The Comentioning

confidence: 96%

Sparse Approximation via Generating Point Sets

Blum

Har-Peled

Raichel

2019

ACM Trans. Algorithms

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For a set P of n points in the unit ball b ⊆ R d , consider the problem of finding a small subset T ⊆ P such that its convex-hull ε-approximates the convex-hull of the original set. Specifically, the Hausdorff distance between the convex hull of T and the convex hull of P should be at most ε. We present an efficient algorithm to compute such an ε -approximation of size k alg , where ε is a function of ε, and k alg is a function of the minimum size k opt of such an ε-approximation. Surprisingly, there is no dependence on the dimension d in either of the bounds. Furthermore, every point of P can be ε-approximated by a convex-combination of points of T that is O(1/ε 2 )-sparse.Our result can be viewed as a method for sparse, convex autoencoding: approximately representing the data in a compact way using sparse combinations of a small subset T of the original data. The new algorithm can be kernelized, and it preserves sparsity in the original input. * A preliminary version of this paper appeared in SODA 16 [BHR16]. The full version of the paper is also available on the arxiv [BHR15].

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“…The following is well known and is included for the sake of completeness, see Reference [13]. It also follows readily from the Preceptron algorithm (see Remark 2.8 below).…”

Section: Algorithm For Approximately Computing the Distance To The Comentioning

confidence: 96%

Sparse Approximation via Generating Point Sets

Blum

Har-Peled

Raichel

2019

ACM Trans. Algorithms

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“…Various models of computation utilizing oracles have been previously studied within the community. Examples of other models include nearest-neighbor oracles (i.e., black-box access to nearest neighbor queries over a point set P ) [HKMR16], and proximity probes (which given a convex polygon C and a query q, returns the distance from q to C) [PAvdSG13]. It is reasonable to ask what classification-type problems can be solved with few oracle queries when using separation oracles.…”

Section: Additional Motivation and Some Previous Workmentioning

confidence: 99%

Active Learning a Convex Body in Low Dimensions

Har-Peled¹,

Jones²,

Rahul³

2019

Preprint

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Consider a set P ⊆ R d of n points, and a convex body C provided via a separation oracle. The task at hand is to decide for each point of P if it is in C using the fewest number of oracle queries. We show that one can solve this problem in two and three dimensions using O( P log n) queries, where P is the largest subset of points of P in convex position. Furthermore, we show that in two dimensions one can solve this problem using O( (P, C) log 2 n) oracle queries, where (P, C) is a lower bound on the minimum number of queries that any algorithm for this specific instance requires.

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“…In general, computational models that involve various oracles as algorithmic building blocks have been studied in computational geometry, as they represent algorithms in which the input is given implicitly, and access to any information provided by the input is done by oracle queries. See Har-Peled et al [HKMR16] and references therein for such examples.…”

Section: Introductionmentioning

confidence: 99%

On Undecided LP, Clustering and Active Learning

Ashur¹,

Har-Peled²

2021

Preprint

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We study colored coverage and clustering problems. Here, we are given a colored point set, where the points are covered by (unknown) k clusters, which are monochromatic (i.e., all the points covered by the same cluster, have the same color). The access the colors of the points (or even the points themselves) is provided indirectly via various queries (such as nearest neighbor, or separation queries). We show that one can correctly deduce the color of all the points (i.e., compute a monochromatic clustering of the points) using a polylogarithmic number of queries, if the number of clusters is a constant.We investigate several variants of this problem, including Undecided Linear Programming, covering of points by k monochromatic balls, covering by k triangles/simplices, and terrain simplification. For the later problem, we present the first near linear time approximation algorithm. While our approximation is slightly worse than previous work, this is the first algorithm to have subquadratic complexity if the terrain has "small" complexity.

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Space Exploration via Proximity Search

Cited by 5 publications

References 21 publications

Sparse Approximation via Generating Point Sets

Sparse Approximation via Generating Point Sets

Active Learning a Convex Body in Low Dimensions

On Undecided LP, Clustering and Active Learning

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