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Metric (mathematics)

About: Metric (mathematics) is a research topic. Over the lifetime, 42617 publications have been published within this topic receiving 836571 citations. The topic is also known as: distance function & metric.


Papers
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Proceedings ArticleDOI
02 Apr 2001
TL;DR: A new distance function D/sub tw-lb/ that consistently underestimates the time warping distance and also satisfies the triangular inequality is devised and achieves significant speedup up to 43 times with real-world S&P 500 stock data and up to 720 times with very large synthetic data.
Abstract: This paper proposes a new novel method for similarity search that supports time warping in large sequence databases. Time warping enables finding sequences with similar patterns even when they are of different lengths. Previous methods for processing similarity search that supports time warping fail to employ multi-dimensional indexes without false dismissal since the time warping distance does not satisfy the triangular inequality. Our primary goal is to innovate on search performance without permitting any false dismissal. To attain this goal, we devise a new distance function D/sub tw-lb/ that consistently underestimates the time warping distance and also satisfies the triangular inequality D/sub tw-lb/ uses a 4-tuple feature vector that is extracted from each sequence and is invariant to time warping. For efficient processing of similarity search, we employ a multi-dimensional index that uses the 4-tuple feature vector as indexing attributes and D/sub tw-lb/ as a distance function. The extensive experimental results reveal that our method achieves significant speedup up to 43 times with real-world S&P 500 stock data and up to 720 times with very large synthetic data.

337 citations

Journal ArticleDOI
Roger N. Shepard1
TL;DR: In this article, the authors used both mathematical and Monte Carlo results to establish and clarify the possibility of extracting metric information from purely ordinal data for two multidimensional cases: (a) analysis of proximities, in which one is given a single rank order of all n(n−1) 2 pairs of n objects with respect to psychological similarity or "proximity".

337 citations

Journal ArticleDOI
TL;DR: The robust sampled problem is shown to be a good approximation for the ambiguous chance constrained problem with a high probability using the Strassen-Dudley Representation Theorem that states that when the distributions of two random variables are close in the Prohorov metric one can construct a coupling of the random variables such that the samples are close with ahigh probability.
Abstract: In this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain. We focus primarily on the special case where the uncertainty set ** of the distributions is of the form ** where ρp denotes the Prohorov metric. The ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure **. The main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability. This result is established using the Strassen-Dudley Representation Theorem that states that when the distributions of two random variables are close in the Prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability. We also show that the robust sampled problem can be solved efficiently both in theory and in practice.

337 citations

Proceedings ArticleDOI
17 May 2008
TL;DR: This work defines an isometry invariant Max Min COV(X) which bounds from below the performance of Lipschitz MAB algorithms for X, and presents an algorithm which comes arbitrarily close to meeting this bound.
Abstract: In a multi-armed bandit problem, an online algorithm chooses from a set of strategies in a sequence of $n$ trials so as to maximize the total payoff of the chosen strategies. While the performance of bandit algorithms with a small finite strategy set is quite well understood, bandit problems with large strategy sets are still a topic of very active investigation, motivated by practical applications such as online auctions and web advertisement. The goal of such research is to identify broad and natural classes of strategy sets and payoff functions which enable the design of efficient solutions. In this work we study a very general setting for the multi-armed bandit problem in which the strategies form a metric space, and the payoff function satisfies a Lipschitz condition with respect to the metric. We refer to this problem as the "Lipschitz MAB problem". We present a complete solution for the multi-armed problem in this setting. That is, for every metric space (L,X) we define an isometry invariant Max Min COV(X) which bounds from below the performance of Lipschitz MAB algorithms for $X$, and we present an algorithm which comes arbitrarily close to meeting this bound. Furthermore, our technique gives even better results for benign payoff functions.

336 citations

Journal ArticleDOI
E V Ruiz1
TL;DR: A new algorithm is proposed which finds the Nearest Neighbour of a given sample in approximately constant average time complexity, independent of the data set size, thus being of general use in many present applications of Pattern Recognition.

335 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202253
20213,191
20203,141
20192,843
20182,731
20172,341