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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: A new method for source localization is described that is based on a modification of the well-known MUSIC algorithm, and a general form of the RAP-MUSIC algorithm is described for the case of diversely polarized sources.
Abstract: A new method for source localization is described that is based on a modification of the well-known MUSIC algorithm. In classical MUSIC, the array manifold vector is projected onto an estimate of the signal subspace. Errors in the estimate of the signal subspace can make localization of multiple sources difficult. Recursively applied and projected (RAP) MUSIC uses each successively located source to form an intermediate array gain matrix and projects both the array manifold and the signal subspace estimate into its orthogonal complement. The MUSIC projection to find the next source is then performed in this reduced subspace. Special assumptions about the array manifold structure, such as Vandermonde or shift invariance, are not required. Using the metric of principal angles, we describe a general form of the RAP-MUSIC algorithm for the case of diversely polarized sources. Through a uniform linear array simulation with two highly correlated sources, we demonstrate the improved Monte Carlo error performance of RAP-MUSIC relative to MUSIC and two other sequential subspace methods: S and IES-MUSIC. We then demonstrate the more general utility of this algorithm for multidimensional array manifolds in a magnetoencephalography (MEG) source localization simulation.

365 citations

Journal ArticleDOI
TL;DR: In this paper, the authors fully extend to the Heisenberg group endowed with its intrinsic Carnot-Caratheodory metric and perimeter the classical De Giorgi's rectifiability divergence theorems.
Abstract: In this paper, we fully extend to the Heisenberg group endowed with its intrinsic Carnot-Caratheodory metric and perimeter the classical De Giorgi's rectifiability divergence theorems.

364 citations

Journal ArticleDOI
TL;DR: In this paper, the geometric interpretation of the expected value and the variance in real Euclidean space is used as a starting point to introduce metric counterparts on an arbitrary finite dimensional Hilbert space.
Abstract: The geometric interpretation of the expected value and the variance in real Euclidean space is used as a starting point to introduce metric counterparts on an arbitrary finite dimensional Hilbert space. This approach allows us to define general reasonable properties for estimators of parameters, like metric unbiasedness and minimum metric variance, resulting in a useful tool to better understand the logratio approach to the statistical analysis of compositional data, who's natural sample space is the simplex.

362 citations

Journal ArticleDOI
TL;DR: The dominant contribution to the path integral seems to come from metrics which have about one unit of topology per Planck volume as discussed by the authors, which can be described by introducing a Λ term as a Lagrange multiplier for the 4 volume.

361 citations

Proceedings ArticleDOI
25 Jul 2004
TL;DR: It is empirically demonstrated that two of the most acclaimed CF recommendation algorithms have flaws that result in a dramatically unacceptable user experience, and a new Belief Distribution Algorithm is introduced that overcomes these flaws and provides substantially richer user modeling.
Abstract: Collaborative Filtering (CF) systems have been researched for over a decade as a tool to deal with information overload. At the heart of these systems are the algorithms which generate the predictions and recommendations.In this article we empirically demonstrate that two of the most acclaimed CF recommendation algorithms have flaws that result in a dramatically unacceptable user experience.In response, we introduce a new Belief Distribution Algorithm that overcomes these flaws and provides substantially richer user modeling. The Belief Distribution Algorithm retains the qualities of nearest-neighbor algorithms which have performed well in the past, yet produces predictions of belief distributions across rating values rather than a point rating value.In addition, we illustrate how the exclusive use of the mean absolute error metric has concealed these flaws for so long, and we propose the use of a modified Precision metric for more accurately evaluating the user experience.

360 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