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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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Book ChapterDOI
05 Sep 2010
TL;DR: This paper proposes to learn a metric using those labeled pairs of bags, leading to MildML, for multiple instance logistic discriminant metric learning, and finds that MildML using the bag-level annotation performs as well as fully supervised metric learning using instance- level annotation.
Abstract: Metric learning aims at finding a distance that approximates a task-specific notion of semantic similarity. Typically, a Mahalanobis distance is learned from pairs of data labeled as being semantically similar or not. In this paper, we learn such metrics in a weakly supervised setting where "bags" of instances are labeled with "bags" of labels. We formulate the problem as a multiple instance learning (MIL) problem over pairs of bags. If two bags share at least one label, we label the pair positive, and negative otherwise. We propose to learn a metric using those labeled pairs of bags, leading to MildML, for multiple instance logistic discriminant metric learning. MildML iterates between updates of the metric and selection of putative positive pairs of examples from positive pairs of bags. To evaluate our approach, we introduce a large and challenging data set, Labeled Yahoo! News, which we have manually annotated and contains 31147 detected faces of 5873 different people in 20071 images. We group the faces detected in an image into a bag, and group the names detected in the caption into a corresponding set of labels. When the labels come from manual annotation, we find that MildML using the bag-level annotation performs as well as fully supervised metric learning using instance-level annotation. We also consider performance in the case of automatically extracted labels for the bags, where some of the bag labels do not correspond to any example in the bag. In this case MildML works substantially better than relying on noisy instance-level annotations derived from the bag-level annotation by resolving face-name associations in images with their captions.

191 citations

Journal ArticleDOI
TL;DR: This paper presents approximation algorithms for median problems in metric spaces and fixed-dimensional Euclidean space that use a new method for transforming an optimal solution of the linear program relaxation of the s-median problem into a provably good integral solution.

191 citations

Journal ArticleDOI
TL;DR: In this article, the authors present the complete family of space-times of type D with an aligned electromagnetic field and a possibly non-zero cosmological constant for the expanding case of the Plebanski-demianski metric.
Abstract: The Plebanski-Demianski metric, and those that can be obtained from it by taking coordinate transformations in certain limits, include the complete family of space-times of type D with an aligned electromagnetic field and a possibly non-zero cosmological constant. Starting with a new form of the line element which is better suited both for physical interpretation and for identifying different subfamilies, we review this entire family of solutions. Our metric for the expanding case explicitly includes two parameters which represent the acceleration of the sources and the twist of the repeated principal null congruences, the twist being directly related to both the angular velocity of the sources and their NUT-like properties. The non-expanding type D solutions are also identified. All special cases are derived in a simple and transparent way.

191 citations

Proceedings ArticleDOI
09 Aug 2003
TL;DR: This work proposes a natural metric on controller parameterization that results from considering the manifold of probability distributions over paths induced by a stochastic controller that leads to a covariant gradient ascent rule.
Abstract: We investigate the problem of non-covariant behavior of policy gradient reinforcement learning algorithms. The policy gradient approach is amenable to analysis by information geometric methods. This leads us to propose a natural metric on controller parameterization that results from considering the manifold of probability distributions over paths induced by a stochastic controller. Investigation of this approach leads to a covariant gradient ascent rule. Interesting properties of this rule are discussed, including its relation with actor-critic style reinforcement learning algorithms. The algorithms discussed here are computationally quite efficient and on some interesting problems lead to dramatic performance improvement over noncovariant rules.

191 citations

Journal ArticleDOI
TL;DR: This work quantifies "locality" and bound the locality of multidimensional space-filling curves, which comes close to achieving optimal locality.
Abstract: A space-filling curve is a linear traversal of a discrete finite multidimensional space. In order for this traversal to be useful in many applications, the curve should preserve "locality". We quantify "locality" and bound the locality of multidimensional space-filling curves. Classic Hilbert space-filling curves come close to achieving optimal locality.

191 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