Capturing the Origin with Random Points: Generalizations of a Putnam Problem

Howard, Ralph W.; Sisson, Paul

doi:10.1080/07468342.1996.11973774

Cited by 4 publications

(2 citation statements)

References 7 publications

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“…Then if and only if and the row vectors are not all contained in some hemisphere [31]. Let indicate the event that vectors lie on the same hemisphere, and will denote the complement of .…”

Section: Appendix C Proof Of Theoremmentioning

confidence: 99%

Adaptive Local Linear Regression With Application to Printer Color Management

Gupta

Garcia

Chin

2008

IEEE Trans. on Image Process.

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Local learning methods, such as local linear regression and nearest neighbor classifiers, base estimates on nearby training samples, neighbors. Usually, the number of neighbors used in estimation is fixed to be a global "optimal" value, chosen by cross validation. This paper proposes adapting the number of neighbors used for estimation to the local geometry of the data, without need for cross validation. The term enclosing neighborhood is introduced to describe a set of neighbors whose convex hull contains the test point when possible. It is proven that enclosing neighborhoods yield bounded estimation variance under some assumptions. Three such enclosing neighborhood definitions are presented: natural neighbors, natural neighbors inclusive, and enclosing k-NN. The effectiveness of these neighborhood definitions with local linear regression is tested for estimating lookup tables for color management. Significant improvements in error metrics are shown, indicating that enclosing neighborhoods may be a promising adaptive neighborhood definition for other local learning tasks as well, depending on the density of training samples.

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“…Then if and only if and the row vectors are not all contained in some hemisphere [31]. Let indicate the event that vectors lie on the same hemisphere, and will denote the complement of .…”

Section: Appendix C Proof Of Theoremmentioning

confidence: 99%

Adaptive Local Linear Regression With Application to Printer Color Management

Gupta

Garcia

Chin

2008

IEEE Trans. on Image Process.

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“…For example, a side of a degenerate quadrilateral (one point inside three others) has a density that is non-Rayleigh.Let A, B, C be independent random Gaussian points in R 2 , all of which have mean vector zero and covariance matrix identity. The triangle ABC is said to capture the origin if (0, 0) is contained in the convex hull of A, B, C. This event occurs with probability 1/4 and a highly attractive proof appears in [1]. We offer an unattractive proof of the same, with the advantage that (0, 0) can be replaced by an arbitrary location (ξ, η), but the results are available only numerically.…”

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confidence: 99%

Capturing, Ordering and Gaussianity in 2D

Finch¹

2016

Preprint

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We collect various facts related loosely to random Gaussian quadrilaterals in the plane. For example, a side of a degenerate quadrilateral (one point inside three others) has a density that is non-Rayleigh.Let A, B, C be independent random Gaussian points in R 2 , all of which have mean vector zero and covariance matrix identity. The triangle ABC is said to capture the origin if (0, 0) is contained in the convex hull of A, B, C. This event occurs with probability 1/4 and a highly attractive proof appears in [1]. We offer an unattractive proof of the same, with the advantage that (0, 0) can be replaced by an arbitrary location (ξ, η), but the results are available only numerically. As far as is known, a closed-form expression for probabilities does not exist.We then review the variance of the median of three points in R 1 and examine how this might be generalized to four points in R 2 . The convex-hull peeling characterization for order statistics (better: data depth) in the plane is usually said to be distributionally intractible [2,3,4,5]. The reason will become rather clear here! Multivariate integrals are written down, but no attempt is made to evaluate them.Finally, we summarize what is known about random Gaussian quadrilaterals in R 2 and give some simulation-based outcomes. Our hope (as always) is that someone else might be able to break the theoretical logjam and provide a rigorous analysis supporting these.

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CHoCC: Convex Hull of Cospherical Circles and Applications to Lattices

Gupta

Kurzeja

et al. 2020

Computer-Aided Design

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Capturing the Origin with Random Points: Generalizations of a Putnam Problem

Cited by 4 publications

References 7 publications

Adaptive Local Linear Regression With Application to Printer Color Management

Adaptive Local Linear Regression With Application to Printer Color Management

Capturing, Ordering and Gaussianity in 2D

CHoCC: Convex Hull of Cospherical Circles and Applications to Lattices

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