Topic
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.
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TL;DR: Both subjective user study and objective evaluation show that Flickr distance is more coherent to human perception than Google distance, which makes it a more reasonable distance metric.
Abstract: This paper presents Flickr distance, which is a novel measurement of the relationship between semantic concepts (objects, scenes) in visual domain. For each concept, a collection of images are obtained from Flickr, based on which the improved latent topic based visual language model is built to capture the visual characteristic of this concept. Then Flickr distance between different concepts is measured by the square root of Jensen-Shannon (JS) divergence between the corresponding visual language models. Comparing with WordNet, Flickr distance is able to handle far more concepts existing on the Web, and it can scale up with the increase of concept vocabularies. Comparing with Google distance, which is generated in textual domain, Flickr distance is more precise for visual domain concepts, as it captures the visual relationship between the concepts instead of their co-occurrence in text search results. Besides, unlike Google distance, Flickr distance satisfies triangular inequality, which makes it a more reasonable distance metric. Both subjective user study and objective evaluation show that Flickr distance is more coherent to human perception than Google distance. We also design several application scenarios, such as concept clustering and image annotation, to demonstrate the effectiveness of this proposed distance in image related applications.
174 citations
TL;DR: An anisotropic adaptive discretization method is presented and how computational efficiency can be increased when applying it to the simulation of cardiovascular flow is demonstrated and a new adaptive approach is proposed which controls the mesh adaptation procedure to maintain structured and graded elements near the wall resulting in a more accurate wall shear stress computation.
174 citations
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TL;DR: In this article, the authors define a notion of a measured length space X having nonnegative N-Ricci curvature, for N finite, or having infinity Ricci curvatures bounded below by K, for K a real number.
Abstract: We define a notion of a measured length space X having nonnegative N-Ricci curvature, for N finite, or having infinity-Ricci curvature bounded below by K, for K a real number. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein space P_2(X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdorff limits. We give geometric and analytic consequences.
174 citations
TL;DR: In this paper, a new approach based on the use of density estimators is proposed for estimating the support of a multivariate density, which is motivated in terms of pattern analysis by Grenander and has interesting connections with detection and clustering.
Abstract: We suggest a new approach, based on the use of density estimators, for the problem of estimating the (compact) support of a multivariate density. This subject (motivated in terms of pattern analysis by Grenander) has interesting connections with detection and clustering. A natural class of density-based estimators is defined. Universal consistency results and convergence rates are established for these estimators, with respect to the usual measure-based metric $d_{\mu}$ between sets. Further convergence rates (with respect to both $d_{\mu}$ and the Hausdorff metric $d_H$) are also obtained under some, fairly intuitive, shape restrictions.
173 citations
173 citations