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.

New parametrization for spherically symmetric black holes in metric theories of gravity

Abstract: We propose a new parametric framework to describe in generic metric theories of gravity the spacetime of spherically symmetric and slowly rotating black holes. In contrast to similar approaches proposed so far, we do not use a Taylor expansion in powers of $M/r$, where $M$ and $r$ are the mass of the black hole and a generic radial coordinate, respectively. Rather, we use a continued-fraction expansion in terms of a compactified radial coordinate. This choice leads to superior convergence properties and allows us to approximate a number of known metric theories with a much smaller set of coefficients. The measure of these coefficients via observations of near-horizon processes can be used to effectively constrain and compare arbitrary metric theories of gravity. Although our attention is here focussed on spherically symmetric black holes, we also discuss how our approach could be extended to rotating black holes.

A new dependency and correlation analysis for features

Abstract: The quality of the data being analyzed is a critical factor that affects the accuracy of data mining algorithms. There are two important aspects of the data quality, one is relevance and the other is data redundancy. The inclusion of irrelevant and redundant features in the data mining model results in poor predictions and high computational overhead. This paper presents an efficient method concerning both the relevance of the features and the pairwise features correlation in order to improve the prediction and accuracy of our data mining algorithm. We introduce a new feature correlation metric Q/sub Y/(X/sub i/,X/sub j/) and feature subset merit measure e(S) to quantify the relevance and the correlation among features with respect to a desired data mining task (e.g., detection of an abnormal behavior in a network service due to network attacks). Our approach takes into consideration not only the dependency among the features, but also their dependency with respect to a given data mining task. Our analysis shows that the correlation relationship among features depends on the decision task and, thus, they display different behaviors as we change the decision task. We applied our data mining approach to network security and validated it using the DARPA KDD99 benchmark data set. Our results show that, using the new decision dependent correlation metric, we can efficiently detect rare network attacks such as User to Root (U2R) and Remote to Local (R2L) attacks. The best reported detection rates for U2R and R2L on the KDD99 data sets were 13.2 percent and 8.4 percent with 0.5 percent false alarm, respectively. For U2R attacks, our approach can achieve a 92.5 percent detection rate with a false alarm of 0.7587 percent. For R2L attacks, our approach can achieve a 92.47 percent detection rate with a false alarm of 8.35 percent.

Energy distance

Abstract: Energy distance is a metric that measures the distance between the distributions of random vectors. Energy distance is zero if and only if the distributions are identical, thus it characterizes equality of distributions and provides a theoretical foundation for statistical inference and analysis. Energy statistics are functions of distances between observations in metric spaces. As a statistic, energy distance can be applied to measure the difference between a sample and a hypothesized distribution or the difference between two or more samples in arbitrary, not necessarily equal dimensions. The name energy is inspired by the close analogy with Newton's gravitational potential energy. Applications include testing independence by distance covariance, goodness-of-fit, nonparametric tests for equality of distributions and extension of analysis of variance, generalizations of clustering algorithms, change point analysis, feature selection, and more. WIREs Comput Stat 2016, 8:27-38. doi: 10.1002/wics.1375

Fine-grained Categorization and Dataset Bootstrapping using Deep Metric Learning with Humans in the Loop

Abstract: Existing fine-grained visual categorization methods often suffer from three challenges: lack of training data, large number of fine-grained categories, and high intraclass vs. low inter-class variance. In this work we propose a generic iterative framework for fine-grained categorization and dataset bootstrapping that handles these three challenges. Using deep metric learning with humans in the loop, we learn a low dimensional feature embedding with anchor points on manifolds for each category. These anchor points capture intra-class variances and remain discriminative between classes. In each round, images with high confidence scores from our model are sent to humans for labeling. By comparing with exemplar images, labelers mark each candidate image as either a "true positive" or a "false positive". True positives are added into our current dataset and false positives are regarded as "hard negatives" for our metric learning model. Then the model is retrained with an expanded dataset and hard negatives for the next round. To demonstrate the effectiveness of the proposed framework, we bootstrap a fine-grained flower dataset with 620 categories from Instagram images. The proposed deep metric learning scheme is evaluated on both our dataset and the CUB-200-2001 Birds dataset. Experimental evaluations show significant performance gain using dataset bootstrapping and demonstrate state-of-the-art results achieved by the proposed deep metric learning methods.