Geometric Discrepancy

Matoušek, Jiřı́

doi:10.1007/978-3-642-03942-3

Cited by 318 publications

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“…Proof As shown in [34], if a halfplane h crosses t edges of the matching, then its discrepancy is O( √ t log n ), with high probability. As shown above, t = O( √ k log(n/k)), for k = |h ∩ P |, so the discrepancy of h is, with high probability,…”

Section: Relative (P ε)-Approximations For Halfplanesmentioning

confidence: 94%

“…In general, range spaces involving semi-algebraic ranges of constant description complexity, i.e., semi-algebraic sets defined as a Boolean combination of a constant number of polynomial equations and inequalities of constant maximum degree, have finite VC-dimension. Halfspaces, balls, ellipsoids, simplices, and boxes are examples of ranges of this kind; see [12,22,34,37] for definitions and more details.…”

Section: Introductionmentioning

confidence: 99%

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Relative (p,ε)-Approximations in Geometry

Har-Peled

Sharir

2010

Discrete Comput Geom

141

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We re-examine the notion of relative (p, ε)-approximations, recently introduced in Cohen et al. (Manuscript, 2006), and establish upper bounds on their size, in general range spaces of finite VC-dimension, using the sampling theory developed in Li et al. (J. Comput. Syst. Sci. 62:516-527, 2001) and in several earlier studies (Pollard in Manuscript, 1986; Haussler in Inf. Comput. 100:78-150, 1992; Talagrand in Ann. Probab. 22:28-76, 1994). We also survey the different notions of sampling, used in computational geometry, learning, and other areas, and show how they relate to each other. We then give constructions of smaller-size relative (p, ε)-approximations for range spaces that involve points and halfspaces in two and higher dimensions. The planar construction is based on a new structure-spanning trees with small relative crossing number, which we believe to be of independent interest. Relative (p, ε)-approximations arise in several geometric problems, such as approximate range counting, and we apply our new structures to obtain efficient solutions for approximate range counting in three dimensions. We also present a simple solution for the planar case.

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Section: Relative (P ε)-Approximations For Halfplanesmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Relative (p,ε)-Approximations in Geometry

Har-Peled

Sharir

2010

Discrete Comput Geom

141

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“…The third category is lattices, which can be considered as a generalization of grids that allows nonorthogonal axes [126,153,165]. As an example, consider Figure 5.5b, which shows 196 lattice points generated by the following technique.…”

Section: Low-discrepancy Samplingmentioning

confidence: 99%

“…A technique for choosing the α parameters by using the roots of irreducible polynomials is discussed in [126]. The ith sample in the lattice is…”

Section: Low-discrepancy Samplingmentioning

confidence: 99%

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Combinatorial Motion Planning

LaValle

2006

Planning Algorithms

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Mathematics of Analog‐to‐Digital Conversion

Güntürk

2012

Comm Pure Appl Math

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As the performance of digital circuits and the capacity of digital communication channels have advanced steadily over the past decades, digital signals have replaced analog signals in nearly every technological application domain, offering robustness, ever increasing quality, and speed. At the same time, many signals that we perceive audiovisually (sound and video), or measure via instrumentation (e.g., geophysical signals and medical images), are analog in their nature. Consequently, not only are the analog‐to‐digital (A/D) and digital‐to‐analog (D/A) conversion processes as crucial as ever, but also the performance of these processes must improve to meet the increasing demands at the digital processing end. Both A/D and D/A conversion is necessarily carried out via analog devices bound by physical limitations such as imprecisions and noise, which present continual technological as well as theoretical challenges to achieving this objective. This article aims to introduce some of the modern mathematical ideas, developments, and challenges in this context from the author's perspective. © 2012 Wiley Periodicals, Inc.

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Geometric Discrepancy

Cited by 318 publications

References 0 publications

Relative (p,ε)-Approximations in Geometry

Relative (p,ε)-Approximations in Geometry

Combinatorial Motion Planning

Mathematics of Analog‐to‐Digital Conversion

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