Asymmetric Metrics Research Articles

Polyhedral compactifications, I

Abstract In this work we describe horofunction compactifications of metric spaces and finite-dimensional real vector spaces through asymmetric metrics and asymmetric polyhedral norms by means of nonstandard methods, that is, by ultrapowers of the spaces at hand. The polyhedral compactifications of the vector spaces carry the structure of stratified spaces with the strata indexed by dual faces of the polyhedral unit ball. Explicit neighborhood bases and descriptions of the horofunctions are provided.

Asymmetrically weighted graphs, asymmetric metrics and large scale geometry

Asymmetrically weighted graphs, asymmetric metrics and large scale geometry

Otimização de portfólio: Métrica do risco com espaço objetivo aumentado

O modelo de portfólio EV eficiente de Markowitz, dado um retorno mínimo requerido, minimiza a variância do portfólio, uma métrica do risco de tendência central calculada pelo método estatístico de concentração de dados, e assim utiliza uma fórmula literal permitindo a solução da otimização por um algoritmo quadrático, exigindo pouco consumo computacional. As evoluções do modelo de Markowitz para métricas assimétricas do risco, minimizam e/ou maximizam o risco, abaixo e/ou acima de um alvo t, como o downside risk, o mean-separated target deviations, o value at risk e o conditional value at risk, porém, não permitem utilizar uma fórmula literal para solução da otimização, transformada em um algoritmo não suave, com solução complexa e maior consumo computacional. O aspecto relevante do modelo de Markowitz foi mostrar que o mais importante não é o risco do ativo mas a contribuição que cada ativo fornece para o risco do portfólio, que depende das interrelações entre os ativos, a covariância do portfólio. Estendendo o raciocínio como contribuição original e relevante, o artigo apresenta uma nova métrica assimétrica do risco, com maior detalhamento das interrelações entre os ativos, aumentando o espaço objetivo da otimização, com maior número de parâmetros otimizados, possibilitando a busca por melhores resultados e utilizando uma expressão literal permitindo solução por um algoritmo não linear, menos complexo que o algoritmo não suave. A análise bibliométrica realizada demonstra a originalidade da evolução do modelo de Markowitz para métricas assimétricas do risco, apresentando fórmula literal para solução e com espaço objetivo aumentado.

Intersection number and some metrics on Teichmüller space

Let $T(X)$ be the Teichmuller space of a closed surface $X$ of genus $g \geq 2,$ $C(X)$ be the space of geodesic currents on $X,$ and $L: T(X) \to C(X)$ be the embedding introduced by Bonahon which maps a hyperbolic metric to its corresponding Liouville current. In this paper, we compare some quantitative relations and topological behaviors between the intersection number and the Teichmuller metric, the length spectrum metric and Thurston's asymmetric metrics on $T(X),$ respectively.

Asymmetric Behavior of Vegetation Seasonal Growth and the Climatic Cause: Evidence from Long-Term NDVI Dataset in Northeast China

Land surface phenology is a response of vegetation to local climate and to climate change, leading to crucial impacts on plant growth rhythm and productivity. Differences in vegetation growth activities in earlier and latter parts of the growing season are tightly correlated to phenological changes and the temporal distribution of plant productivity. However, its spatiotemporal pattern and climatic constraints are poorly understood. For Northeast China (NEC), long-term remotely-sensed vegetation greenness records (NDVI) were employed to quantify seasonally asymmetrical characteristics of vegetation growth in detail, which consists of asymmetry in growing rate (AsyR), mean vegetation greenness (AsyV), and growing period length (AsyL) during vegetation green up and senescence stages (simply termed as spring and autumn). Furthermore, the impact of temperature and precipitation on these indices were examined using relative importance analysis. The results indicate these asymmetric metrics present a pronounced interannual variability profile with a potential cycle of ten years (significant in AsyV and AsyR) for the entire NEC. AsyV is changing synchronously with AsyL but asynchronously with AsyR. The geographical distribution of asymmetric indices shows a similar pattern to identified vegetation cover types, especially in distinguishing crops from natural vegetation. Spatial-averaged asymmetric indices indicate spring production is greater than autumn production (reflected by negative AsyV) across most vegetation types in NEC, yet autumn is longer than spring in all vegetation types, which is identified by positive AsyL. Negative AsyR is mainly found in forests implying there is rapid green up and slow senescence in trees. From a temporal perspective, AsyV decreases with time in forested regions but increases in cropland and grassland, which is similar to the pattern for AsyL. AsyR primarily exhibits a positive trend in forest and a negative trend in cropland and grassland. A relative importance analysis indicates that asymmetries of temperature (AsyTemp) and precipitation (AsyPrcp) play an equal role in significantly affecting vegetation asymmetries in greenness and growth rate but are insignificant to growing season length. AsyTemp mainly presents an obvious contribution to changes in AsyR and AsyV over cropland and grassland. AsyPrcp shows a more widespread controlling effect on AsyR and AsyV over the NEC, except in eastern broad-leaved forest. For the entire NEC, asymmetries of temperature and precipitation are negatively correlated with AsyR but are positively correlated with AsyV and AsyL. This finding may imply that a warmer (positive AsyTemp) autumn tends to improve the length and intensity of vegetation activity. Thus, the long-term change in vegetation growth asymmetries may provide insights for the altering functions of ecosystems and provide information to more accurately build plant growth models in the context of global climate change. Additionally, when combined with other information, asymmetric indices can serve as a supporting tool in classification of vegetation types.

A fusion probability matrix factorization framework for link prediction

Link prediction is a fundamental research problem in network data analysis. Networks usually contain rich node-to-node topological metrics and their effective use is crucial to solve the link prediction problem. Despite significant advances, the existing metric-based link prediction methods usually only consider one single topological metric and thus show some limitations in different types of networks; the existing matrix factorization-based models mainly focus on modeling the adjacent matrix of a network, and this is hard to ensure the modeling of those topological metrics that can play an important role in link prediction. This study develops effective approaches by fusing the adjacent matrix and some key topological metrics in a unified probability matrix factorization framework. In these approaches, we consider not only the symmetric metrics but also the asymmetric metrics which are usually not taken into consideration in the related work. In our probability matrix factorization framework, we first present two fusion models by fusing two kinds of metrics respectively, and based on the fusion models, we put forward the final fusion models which fuse the two kinds of metrics simultaneously. To verify the performance of all the fusion models, we conduct the experiments with six directed networks and six undirected ones, and the extensive experiments show that the proposed models provide impressive predicting performance for link prediction.

Fuzzy Random Walkers with Second Order Bounds: An Asymmetric Analysis

Edge-fuzzy graphs constitute an essential modeling paradigm across a broad spectrum of domains ranging from artificial intelligence to computational neuroscience and social network analysis. Under this model, fundamental graph properties such as edge length and graph diameter become stochastic and as such they are consequently expressed in probabilistic terms. Thus, algorithms for fuzzy graph analysis must rely on non-deterministic design principles. One such principle is Random Walker, which is based on a virtual entity and selects either edges or, like in this case, vertices of a fuzzy graph to visit. This allows the estimation of global graph properties through a long sequence of local decisions, making it a viable strategy candidate for graph processing software relying on native graph databases such as Neo4j. As a concrete example, Chebyshev Walktrap, a heuristic fuzzy community discovery algorithm relying on second order statistics and on the teleportation of the Random Walker, is proposed and its performance, expressed in terms of community coherence and number of vertex visits, is compared to the previously proposed algorithms of Markov Walktrap, Fuzzy Walktrap, and Fuzzy Newman–Girvan. In order to facilitate this comparison, a metric based on the asymmetric metrics of Tversky index and Kullback–Leibler divergence is used.

Energy, Hopf Differential, and Metrics on Teichmüller Space

Let \(T(X)\) be the Teichmuller space of a closed surface \(X\) with genus \(g \ge 2\). For a given point \([\sigma ] \in T(X)\), let \(Q(\sigma )\) be the space of holomorphic quadratic differentials on \((X, \sigma )\). Denote by \(\Phi : T(X) \rightarrow Q(\sigma )\) the Hopf mapping which maps a point \([\rho ] \in T(X)\) to the Hopf differential of the unique harmonic map \(w: (X, \sigma ) \rightarrow (X, \rho )\) in the homotopy class of the identity \(id: (X, \sigma ) \rightarrow (X, \rho )\). In this paper, we give quantitative comparisons of both the energy and \(L_1\)-norm of the Hopf differential involved in \(\Phi \) with the length spectrum metric, the Teichmuller metric, and Thurston’s asymmetric metrics, respectively.

SOME REMARKS ON THURSTON METRIC AND HYPERBOLIC METRIC

In this paper, we study the relations between the Thurston metric and the hyperbolic metric on a closed surface of genus . We show a rigidity result which says if there is an inequality between the marked length spectra of these two metrics, then they are isotopic. We obtain some inequalities on length comparisons between these metrics. Besides, we show certain distance distortions under conformal graftings, with respect to the metric, the length spectrum metric and Thurston`s asymmetric metrics.

The Directed Orienteering Problem

This paper studies vehicle routing problems on asymmetric metrics. Our starting point is the directed k-TSP problem: given an asymmetric metric (V,d), a root r∈V and a target k≤|V|, compute the minimum length tour that contains r and at least k other vertices. We present a polynomial time $O(\frac{\log^{2} n}{\log\log n}\cdot\log k)$ -approximation algorithm for this problem. We use this algorithm for directed k-TSP to obtain an $O(\frac{\log^{2} n}{\log\log n})$ -approximation algorithm for the directed orienteering problem. This answers positively, the question of poly-logarithmic approximability of directed orienteering, an open problem from Blum et al. (SIAM J. Comput. 37(2):653–670, 2007). The previously best known results were quasi-polynomial time algorithms with approximation guarantees of O(log 2 k) for directed k-TSP, and O(log n) for directed orienteering (Chekuri and Pal in IEEE Symposium on Foundations in Computer Science, pp. 245–253, 2005). Using the algorithm for directed orienteering within the framework of Blum et al. (SIAM J. Comput. 37(2):653–670, 2007) and Bansal et al. (ACM Symposium on Theory of Computing, pp. 166–174, 2004), we also obtain poly-logarithmic approximation algorithms for the directed versions of discounted-reward TSP and vehicle routing problem with time-windows.

Evaluation of novelty metrics for sentence-level novelty mining

This work addresses the problem of detecting novel sentences from an incoming stream of text data, by studying the performance of different novelty metrics, and proposing a mixed metric that is able to adapt to different performance requirements. Existing novelty metrics can be divided into two types, symmetric and asymmetric, based on whether the ordering of sentences is taken into account. After a comparative study of several different novelty metrics, we observe complementary behavior in the two types of metrics. This finding motivates a new framework of novelty measurement, i.e. the mixture of both symmetric and asymmetric metrics. This new framework of novelty measurement performs superiorly under different performance requirements varying from high-precision to high-recall as well as for data with different percentages of novel sentences. Because it does not require any prior information, the new metric is very suitable for real-time knowledge base applications such as novelty mining systems where no training data is available beforehand.