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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, it was shown that in certain compactifications of Minkowski space-time theory on eight-manifolds, the four-form field strength can have a non-vanishing expectation value, while an $N=2$ supersymmetry is preserved.
Abstract: We show that in certain compactifications of ${\cal M}$-theory on eight-manifolds to three-dimensional Minkowski space-time the four-form field strength can have a non-vanishing expectation value, while an $N=2$ supersymmetry is preserved. For these compactifications a warp factor for the metric has to be taken into account. This warp factor is non-trivial in three space-time dimensions due to Chern-Simons corrections to the fivebrane Bianchi identity. While the original metric on the internal space is not K\"ahler, it can be conformally transformed to a metric that is K\"ahler and Ricci flat, so that the internal manifold has $SU(4)$ holonomy.

466 citations

Proceedings ArticleDOI
03 Apr 2017
TL;DR: The proposed algorithm outperforms state-of-the-art collaborative filtering algorithms on a wide range of recommendation tasks and uncovers the underlying spectrum of users' fine-grained preferences.
Abstract: Metric learning algorithms produce distance metrics that capture the important relationships among data. In this work, we study the connection between metric learning and collaborative filtering. We propose Collaborative Metric Learning (CML) which learns a joint metric space to encode not only users' preferences but also the user-user and item-item similarity. The proposed algorithm outperforms state-of-the-art collaborative filtering algorithms on a wide range of recommendation tasks and uncovers the underlying spectrum of users' fine-grained preferences. CML also achieves significant speedup for Top-K recommendation tasks using off-the-shelf, approximate nearest-neighbor search, with negligible accuracy reduction.

465 citations

Journal ArticleDOI
TL;DR: A multi-resolution image fusion metric using visual information fidelity (VIF) is presented to assess fusion performance objectively and is found that VIFF performs better in terms of both human perception matching and computational complexity.

462 citations

Posted Content
TL;DR: New insights are provided into the Inception Score, a recently proposed and widely used evaluation metric for generative models, and it is demonstrated that it fails to provide useful guidance when comparing models.
Abstract: Deep generative models are powerful tools that have produced impressive results in recent years. These advances have been for the most part empirically driven, making it essential that we use high quality evaluation metrics. In this paper, we provide new insights into the Inception Score, a recently proposed and widely used evaluation metric for generative models, and demonstrate that it fails to provide useful guidance when comparing models. We discuss both suboptimalities of the metric itself and issues with its application. Finally, we call for researchers to be more systematic and careful when evaluating and comparing generative models, as the advancement of the field depends upon it.

459 citations

Journal ArticleDOI
TL;DR: A principal-components procedure was employed to reduce simple multicollinear complexity metrics to uncorrelated measures on orthogonal complexity domains to classify programs into alternate groups, depending on the metric values of the program.
Abstract: The use of the statistical technique of discriminant analysis as a tool for the detection of fault-prone programs is explored. A principal-components procedure was employed to reduce simple multicollinear complexity metrics to uncorrelated measures on orthogonal complexity domains. These uncorrelated measures were then used to classify programs into alternate groups, depending on the metric values of the program. The criterion variable for group determination was a quality measure of faults or changes made to the programs. The discriminant analysis was conducted on two distinct data sets from large commercial systems. The basic discriminant model was constructed from deliberately biased data to magnify differences in metric values between the discriminant groups. The technique was successful in classifying programs with a relatively low error rate. While the use of linear regression models has produced models of limited value, this procedure shows great promise for use in the detection of program modules with potential for faults. >

458 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