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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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Journal ArticleDOI
TL;DR: In this paper, the authors prove equivalent conditions for two-sided sub-Gaussian estimates of heat kernels on metric measure spaces and show that these are equivalent to the conditions for heat kernels in the case of two sides.
Abstract: We prove equivalent conditions for two-sided sub-Gaussian estimates of heat kernels on metric measure spaces.

156 citations

Journal ArticleDOI
Kentaro Toyama1, Andrew Blake1
TL;DR: A new, exemplar-based, probabilistic paradigm for visual tracking is presented, which provides alternatives to standard learning algorithms by allowing the use of metrics that are not embedded in a vector space and uses a noise model that is learned from training data.
Abstract: A new, exemplar-based, probabilistic paradigm for visual tracking is presented. Probabilistic mechanisms are attractive because they handle fusion of information, especially temporal fusion, in a principled manner. Exemplars are selected representatives of raw training data, used here to represent probabilistic mixture distributions of object configurations. Their use avoids tedious hand-construction of object models, and problems with changes of topology. Using exemplars in place of a parameterized model poses several challenges, addressed here with what we call the “Metric Mixture” (M2) approach, which has a number of attractions. Principally, it provides alternatives to standard learning algorithms by allowing the use of metrics that are not embedded in a vector space. Secondly, it uses a noise model that is learned from training data. Lastly, it eliminates any need for an assumption of probabilistic pixelwise independence. Experiments demonstrate the effectiveness of the M2 model in two domains: tracking walking people using “chamfer” distances on binary edge images, and tracking mouth movements by means of a shuffle distance.

156 citations

Journal ArticleDOI
TL;DR: In this article, the authors discuss convergence of pointed metric measure spaces in absence of any compactness condition and show that all of them are equivalent and that for doubling spaces these are also equivalent to the measured-Gromov-Hausdorff convergence.
Abstract: Aim of this paper is to discuss convergence of pointed metric measure spaces in absence of any compactness condition. We propose various definitions, show that all of them are equivalent and that for doubling spaces these are also equivalent to the well known measured-Gromov-Hausdorff convergence. Then we show that the curvature conditions CD(K,∞) and RCD(K,∞) are stable under this notion of convergence and that the heat flow passes to the limit as well, both in the Wasserstein and in the L-framework. We also prove the variational convergence of Cheeger energies in the naturally adapted Γ-Mosco sense and the convergence of the spectra of the Laplacian in the case of spaces either uniformly bounded or satisfying the RCD(K,∞) condition with K > 0. When applied to Riemannian manifolds, our results allow for sequences with diverging dimensions.

156 citations

Patent
Dennis Hui1, Johan Axnäs1, Kumar Balachandran1, Robert Baldemair1, Hugo Tullberg1 
24 Sep 2013
TL;DR: In this article, a link metric is computed for each of the one or more hypothesized transmission modes for the link and at least one of the link metrics are then provided to a routing update module.
Abstract: Systems and methods for computing and/or utilizing mutual information based link metrics for a link in a wireless mesh network are disclosed. In one embodiment, one or more mutual information values are computed for a link between a transmitter of a first network node and a receiver of a second network node in a wireless mesh network. Each of the one or more mutual information values is computed for a different hypothesized transmission mode for the link. One or more link metrics for the link are computed as a function of the mutual information values, where each link metric is computed based on a different one of the one or more mutual information values. In this manner, a link metric is computed for each of the one or more hypothesized transmission modes for the link. At least one of the link metrics are then provided to a routing update module.

156 citations

Proceedings ArticleDOI
30 Aug 2015
TL;DR: This paper identifies categories of data sets were as few as 50 instances are enough to build a defect prediction model, and shows that empirically and theoretically, “large enough” may be very small indeed.
Abstract: Software defect prediction is one of the most active research areas in software engineering. We can build a prediction model with defect data collected from a software project and predict defects in the same project, i.e. within-project defect prediction (WPDP). Researchers also proposed cross-project defect prediction (CPDP) to predict defects for new projects lacking in defect data by using prediction models built by other projects. In recent studies, CPDP is proved to be feasible. However, CPDP requires projects that have the same metric set, meaning the metric sets should be identical between projects. As a result, current techniques for CPDP are difficult to apply across projects with heterogeneous metric sets. To address the limitation, we propose heterogeneous defect prediction (HDP) to predict defects across projects with heterogeneous metric sets. Our HDP approach conducts metric selection and metric matching to build a prediction model between projects with heterogeneous metric sets. Our empirical study on 28 subjects shows that about 68% of predictions using our approach outperform or are comparable to WPDP with statistical significance.

156 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