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The sliced Wasserstein and more recently max-sliced Wasserstein metrics $\mW_p$ have attracted abundant attention in data sciences and machine learning due to its advantages to tackle the curse of dimensionality. A question of particular importance is the strong equivalence between these projected Wasserstein metrics and the (classical) Wasserstein metric $\Wc_p$. Recently, Paty and Cuturi have proved the strong equivalence of $\mW_2$ and $\Wc_2$. We show that the strong equivalence also holds for $p=1$, while we show that the sliced Wasserstein metric does not share this nice property., Comment: To appear in Electronic Communications in Probability

We consider the recently found connection between geodesically equivalent metrics and integrable geodesic flows. If two different metrics on a manifold have the same geodesics, then the geodesic flows of these metrics admit sufficiently many integrals (of a special form) in involution, and vice versa. The quantum version of this result is also true: if two metrics on one manifold have the same geodesics, then the Beltrami Laplace operator Δ for each metric admits sufficiently many linear differential operators communiting with Δ. This implies that the topology of a manifold with two different metrics with the same geodesics must be sufficiently simple. We also have that the nonproportionality of the metrics at a point implies the nonproportionality of the metrics at almost all points.

We review the solution to the equivalence problem in general relativity given by Cartan and Brans and present a practically useful method to obtain a coordinate-invariant description of a geometry. The method, which can be seen as a generalized Petrov classification, automatically gives the dimensions of the isometry group and its isotropy subgroup. Finally, we illustrate the method using the Schwarzschild solution as a very simple example.

Management of Earth's surface albedo is increasingly viewed as an important climate change mitigation strategy both on (Seneviratne et al., 2018) and off (Field et al., 2018; Kravitz et al., 2018) the land. Assessing the impact of a surface albedo change involves employing a measure like radiative forcing (RF) which can be challenging to digest for decision-makers who deal in the currency of CO2-equivalent emissions. As a result, many researchers express albedo change (Δα) RFs in terms of their CO2-equivalent effects, despite the lack of a standard method for doing so, such as there is for emissions of well-mixed greenhouse gases (WMGHGs; e.g., IPCC AR5, Myhre et al. (2013)). A major challenge for converting Δα RFs into their CO2-equivalant effects in a manner consistent with current IPCC emission metric approaches stems from the lack of a universal time-dependency following the perturbation (perturbation lifetime). Here, we review existing methodologies based on the RF concept with the goal of highlighting the context(s) in which the resulting CO2-equivalent metrics may or may not have merit. To our knowledge this is the first review dedicated entirely to the topic since the first CO2-eq. metric for Δα surfaced 20 years ago. We find that, although there are some methods that sufficiently address the time-dependency issue, none address or sufficiently account for the spatial disparity between the climate response to CO2 emissions and Δα – a major critique of Δα metrics based on the RF concept (Jones et al., 2013). We conclude that considerable research efforts are needed to build consensus surrounding the RF efficacy of various surface forcing types associated with Δα (e.g., crop change, forest harvest, etc.), and the degree to which these are sensitive to the spatial pattern, extent, and magnitude of the underlying surface forcings.

The Bochner tensor is the Kahler analogue of the conformal Weyl tensor. In this article, we derive local (i.e., in a neighbourhood of almost every point) normal forms for a (pseudo-)Kahler manifold with vanishing Bochner tensor. The description is pined down to a new class of symmetric spaces which we describe in terms of their curvature operators. We also give a local description of weakly Bochner-flat metrics defined by the property that the Bochner tensor has vanishing divergence. Our results are based on the local normal forms for c-projectively equivalent metrics. As a by-product, we also describe all Kahler-Einstein metrics admitting a c-projectively equivalent one.

The paper is devoted to maps on metric spaces whose quasiconformal characteristic satisfies certain restrictions of integral type. First of all, we prove that the so-called ring Q-mappings ...

We define metrics on Culler-Vogtmann space, which are an analogue of the Teichmuller metric and are constructed using stretching factors. In fact the metrics we study are related, one being a symmetrised version of the other. We investigate the basic properties of these metrics, showing the advantages and pathologies of both choices. We show how to compute stretching factors between marked metric graphs in an easy way and we discuss the behaviour of stretching factors under iterations of automorphisms. We study metric properties of folding paths, showing that they are geodesic for the non-symmetric metric and, if they do not enter the thin part of Outer space, quasi-geodesic for the symmetric metric., changelog v1 -> v2: Added an example, suggested by Bert Wiest and Thierry Coulbois showing that the symmetric metric is not geodesic. Minor changes. Biblio updated

A family of integrable geodesic flows is obtained. Any such a family corresponds to a pair of geodesically equivalent metrics.

This article studies the relationship between two metrics, the area under the modulation transfer function (MTFa) and the energy efficiency (EE), and their ability to predict the visual quality of patients implanted with multifocal intraocular lenses (IOLs). The optical quality of IOLs is assessed in vitro using two metrics, the MTFa and EE. We measured them for three different multifocal IOLs with parabolic phase profile using image formation, through-focus (TF) scanning, three R, G, B wavelengths, and two pupils. We analyzed the correlation between MTFa and EE. In parallel, clinical defocus curves of visual acuity (VA) were measured and averaged from sets of patients implanted with the same IOLs. An excellent linear correlation was found between the MTFa and EE for the considered IOLs, wavelengths and pupils (R2 > 0.9). We computed the polychromatic TF-MTFa, TF-EE, and derived mathematical relationships between each metrics and clinical average VA. MTFa and EE proved to be equivalent metrics to characterize the optical quality of the studied multifocal IOLs and also in terms of clinical VA predictability. Ministerio de Economía y Competitividad y fondos Feder (DPI2016-76019-R).

Tree metrics are encountered throughout pure and applied mathematics. Their simple structure makes them a convenient choice of metric in many applications from machine learning and computer science. At the same time, there is an elegant theory of harmonic analysis with respect to tree metrics that parallels the classical theory. A basic question in this field, which is of both theoretical and practical interest, is how to design efficient algorithms for building trees with good metric properties. In particular, given a finite metric space, we seek a random family of dominating tree metrics approximating the underlying metric in expectation. For general metrics, this problem has been solved: on the one hand, there are finite metric spaces that cannot be approximated by trees without incurring a distortion logarithmic in the size of the space, while the tree construction of Fakcharoenphol, Rao, and Talwar (FRT, 2003) shows how to achieve such a logarithmic error for arbitrary metrics. Since a distortion that grows even logarithmically with the size of the set may be too large for practical use in many settings, one naturally asks if there is a more restricted class of metrics where one can do better. The main result of this paper is that certain random family of trees already studied in the computer science literature, including the FRT trees, can be used to approximate snowflake metrics (metrics raised to a power less than 1) with expected distortion bounded by its doubling dimension and the degree of snowflaking. We also show that without snowflaking, the metric distortion can be bounded by a term logarithmic in the distance being approximated and linear in the dimension. We also present an optimal algorithm for building a single FRT tree, whose running time is bounded independently of all problem parameters other than the number of points. We conclude by demonstrating our theoretical results on a numerical example, and applying them to the approximation of the Earth Mover's Distance between probability distributions.

We prove that any pair of reasonable cross norms defined on the tensor product of n Banach spaces induce (2k)n−1-Lipschitz equivalent metrics (thus, a unique topology) on the set SX1,…,Xnk of vecto...

This paper develops a theory of harmonic analysis on spaces with tree metrics, extending previous work in this direction by Gavish, Nadler and Coifman (2010) [30] and Gavish and Coifman (2011, 2012) [28] , [29] . We show how a natural system of martingales and martingale differences induced by a partition tree leads to simple and effective characterizations of the Lipschitz norm and its dual for functions on a single tree metric space. The restrictions we place on the tree metrics are far more general than those considered in previous work. As the dual norm is equal to the Earth Mover's Distance (EMD) between two probability distributions, we recover a simple formula for EMD with respect to tree distances presented by Charikar (2002) [36] . We also consider the situation where an arbitrary metric is approximated by the average of a family of dominating tree metrics. We show that the Lipschitz norm and its dual for the tree metrics can be combined to yield an approximation to the corresponding norms for the underlying metric. The main contributions of this paper, however, are the generalizations of the aforementioned results to the setting of the product of two or more tree metric spaces. For functions on a product space, the notion of regularity we consider is not the Lipschitz condition, but rather the mixed Lipschitz condition that controls the size of a function's mixed difference quotient. This condition is extremely natural for datasets that can be described as a product of metric spaces, such as word-document databases. We develop effective formulas for norms equivalent to the mixed Lipschitz norm and its dual, and extend our results on combining pairs of trees.

Let $X,Y$ be Banach spaces, and let $T$ , $\delta T:X\to Y$ be bounded linear operators. Put $\bar{T}=T+\delta T$ . In this article, utilizing the gap between closed subspaces and the perturbation bounds of metric projections, we first present some error estimates of the upper bound of $\Vert \bar{T}^{M}-T^{M}\Vert $ in $L^{p}$ ( $1\lt p\lt +\infty$ ) spaces. Then, by using the concept of strong uniqueness and modulus of convexity, we further investigate the corresponding perturbation bound $\Vert \bar{T}^{M}-T^{M}\Vert $ in uniformly convex Banach spaces.

Some stronger and equivalent metrics are defined on $\mathcal{M}$, the set of all bounded normal operators on a Hilbert space $H$ and then some topological properties of $\mathcal{M}$ are investigated.

The paper provides a generalization of a metric with values in a general structure. Quantales and special kinds of ordered monoids are shown to be suitable for this purpose. Topological and categorical properties of spaces with such generalized metrics are studied and a special emphasis is given on paths in these spaces.

The aim of this paper is to prove some results on the existence and uniqueness of elements of best approximation and continuity of the metric projection in metric spaces. For a subset M of a metric space (X; d), the nature of set of those points of X which have at most one best approximation in M has been discussed. Some equivalent conditions under which an M-space is strictly convex have also been given in this paper.

We study optimal approximations of sets in various metric spaces with sets of balls of equal radius. We consider an Euclidean plane, a sphere, and a plane with a special non-uniform metric. The main component in our constructions of coverings are optimal Chebyshev n-networks and their generalizations. We propose algorithms for constructing optimal coverings based on partitioning a given set into subsets and finding their Chebyshev centers in the Euclidean metric and their counterparts in non-Euclidean ones. Our results have both theoretical and practical value and can be used to solve problems arising in security, communication, and infrastructural logistics.

[EN] In the context of fuzzy metrics in the sense of George and Veeramani, we introduce the concept of stratified fuzzy metric. Many well-known fuzzy metrics are stratified. We prove that stratified strong fuzzy metric spaces (X, M, *) are completable, under the assumption that * is integral (positive). In particular, stratified fuzzy ultrametric spaces are completable. (C) 2017 Elsevier B.V. All rights reserved., Valentin Gregori acknowledges the support of the Spanish Ministry of Economy and Competitiveness under Grant MTM2015-64373-P.

A pseudo-Finsleroid metric function F of the two-axes structure that involves the vertical axis and the horizontal axis is proposed assuming constancy of the curvature of indicatrix. The curvature is negative and the signature of the Finslerian metric tensor is exactly $$(+-\cdots )$$ . The function F endows the tangent space with the geometry which possesses many interesting Finslerian properties. The use of the angle representation is the underlying method which has been conveniently and successfully applied. The appearance of the positive-definite Finsleroid metric function in the horizontal sections of the tangent space is established.

In this paper, we study a class of Finsler metrics, called generalized Douglas–Weyl (GDW) metrics, which includes Douglas metrics and Weyl metrics. We find a sufficient and necessary condition for an orthogonally invariant Finsler metric to be a GDW-metric. As its application, we show that a certain class of Finsler metrics with orthogonal invariance are Douglas metrics if they are GDW-metrics, generalized a theorem previously only known in the case of Weyl metrics.

Under the assumption that $F$ is a strongly convex weakly Kahler Finsler metric on a complex manifold $M$, we prove that $F$ is a weakly complex Berwald metric if and only if $F$ is a real Landsberg metric. This result together with Zhong (2011) implies that among the strongly convex weakly Kahler Finsler metrics there does not exist unicorn metric in the sense of Bao (2007). We also give an explicit example of strongly convex Kahler Finsler metric which is simultaneously a complex Berwald metric, a complex Landsberg metric, a real Berwald metric, and a real Landsberg metric.

The Gromov---Hausdorff distance of two compact metric spaces is a measure for how far the spaces are from being isometric and has been extensively studied in the field of metric geometry. In recent years, a lot of attention has been devoted to computational aspects of the Gromov---Hausdorff distance. One of the most prominent applications is the problem of shape matching, where the goal is to decide whether two shapes given as polygonal meshes are equivalent under certain invariances. Therefore, many methods have been proposed which heuristically estimate the Gromov---Hausdorff distance of metric spaces induced by the shapes. However, the computational complexity of computing the Gromov---Hausdorff distance has not yet been thoroughly investigated. We show that--under standard complexity theoretic assumptions--determining the Gromov---Hausdorff distance of two finite metric spaces cannot be approximated within any reasonable bound in polynomial time. Furthermore, we discover attributes of metric spaces which have a major impact on the complexity of an instance. This enables us to develop an approximation algorithm which is fixed parameter tractable with respect to corresponding parameters.

Over the past decades, distance metric learning has attracted a lot of interest in machine learning and related fields. In this work, we propose an optimization framework for distance metric learning via linear transformations by maximizing the Jeffrey divergence between two multivariate Gaussian distributions derived from local pairwise constraints. In our method, the distance metric is trained on positive and negative difference spaces, which are built from the neighborhood of each training instance, so that the local discriminative information is preserved. We show how to solve this problem with a closed-form solution rather than using tedious optimization procedures. The solution is easy to implement, and tractable for large-scale problems. Experimental results are presented for both a linear and a kernelized version of the proposed method for k-nearest neighbors classification. We obtain classification accuracies superior to the state-of-the-art distance metric learning methods in several cases while being competitive in others. HighlightsWe propose a novel distance metric learning method (DMLMJ) for classification.DMLMJ is simple to implement and it can be solved analytically.We extend DMLMJ into a kernelized version to tackle non-linear problems.Experiments on several data sets show the effectiveness of the proposed method.

In this paper we study properties of weighted Zolotarev metrics and compare them with the Kantorovich metric.

We study the practical behavior of different algorithms and methods that aim to estimate the intrinsic dimension (IDim) in metric spaces. Some of them were specifically developed to evaluate the complexity of searching in metric spaces, based on different theories related to the distribution of distances between objects on such spaces. Others were originally designed for vector spaces only, and have been extended to general metric spaces. To empirically evaluate the fitness of various IDim estimations with the actual difficulty of searching in metric spaces, we compare two representatives of each of the broadest families of metric indices: those based on pivots and those based on compact partitions. Our conclusions are that the estimators Distance Exponent and Correlation fit best their purpose. HighlightsWe study various methods to estimate intrinsic dimensionality.We adapt the methods to the case of metric spaces.We experimentally evaluate how they predict the search effort in real world spaces.Distance Exponent and Correlation are the methods that best predict search difficulty.

We propose an intuitive theorem which uses some concepts of auxiliary functions for establishing existence and uniqueness of the fixed point of a self-mapping. First we work in the setting of fuzzy metric spaces in the sense of George and Veeramani, then we deduce some consequences in modular metric spaces. Finally, a sample homotopy result is derived making use of the main theorem.

[EN] In this paper we introduce and study the concept of strong convergence in fuzzy metric spaces (X, M,*) in the sense of George and Veeramani. This concept is related with the condition Lambda M-t > 0(x, y, t) > 0, which frequently is required or missing in this context. Among other results we characterize the class of s-fuzzy metrics by the strong convergence defined here and we solve partially the question of finding explicitly a compatible metric with a given fuzzy metric., Valentn Gregori acknowledges the support of the Spanish Ministry of Economy and Competitiveness under Grant MTM2015-64373-P (MINECO/FEDER, UE).

Let $(X,d,\preceq )$ be a partially ordered metric space and $f,F$ be single and set valued mappings on $X$. We obtained sufficient conditions for existence of fixed point of mappings $f$ and $F$ on $X$ with a metric transform.

Distance metric plays a significant role in machine learning methods(classification, clustering, etc.), especially in k-nearest neighbor classification(kNN), where the Euclidean distances are computed to decide the labels of unknown points. But Euclidean distance ignores the statistical structure which may help to measure the similarity of different inputs better. In this paper, we construct an unified framework, including two eigenvalue related methods, to learn data-dependent metric. Both methods aim to maximize the difference of intra-class distance and inter-class distance, but the optimization is considered in global view and local view respectively. Different from previous work in metric learning, our methods straight seek for equilibrium between inter-class distance and intra-class distance, and the linear transformation decomposed from the metric is to be optimized directly instead of the metric. Then we can effectively adjust the data distribution in transformed space and construct favorable regions for kNN classification. The problems can be solved simply by eigenvalue-decomposition, much faster than semi-definite programming. After selecting the top eigenvalues, the original data can be projected into low dimensional space, and then insignificant information will be mitigated or eliminated to make the classification more efficiently. This makes it possible that our novel methods make metric learning and dimension reduction simultaneously. The numerical experiments from different points of view verify that our methods can improve the accuracy of kNN classification and make dimension reduction with competitive performance.

In this paper, we introduced soft metric on soft sets and considered its properties. Continuity of soft mappings on soft metric spaces studied. Moreover, we prove the Banach contraction theorem on complete soft metric spaces.

For a metric space ( X , d ) , a subset A resolves ( X , d ) if each point x ∈ X is uniquely determined by the distances d ( x , a ) for a ∈ A . Also the metric dimension of ( X , d ) is the smallest cardinality m d ( X ) such that there is a set A of the cardinality m d ( X ) that resolves X. In this note we are going to determine the metric dimension of metric orbit spaces in some special cases and find an upper bound for a general case. This category contains a vast domain of topological spaces and topological manifolds.

This paper explores 4D seismic history matching and it specifically focuses on the objective function used during the optimisation with seismic data. The objective function is calculated by using binary maps, where one map is obtained from the observed seismic data and the other is from one realisation of the optimisation algorithm from the simulation model. In order to decide which set of parameters is a relevant update for the simulation model, an efficient way is required to measure how similar these two binary images are, during their evaluation within the objective function. Behind this aspect of quantification of the similarities or dissimilarities lies the metric notion, or the art of measuring distances. Four metrics are proposed with this study, the well-known Hamming distance, two widely used metrics, the Hausdorff distance and Mutual Information and a recent metric, called the Current Measure Metric. These metrics will be tested and compared on different case scenarios, designed in accordance to a real field case (gas exsolution) before being used in the second part of the paper. Despite its simplicity, the Hamming distance gives positive results, but the Current Measure Metric appears to be a more efficient choice to cover a wider range of scenarios, these conclusions remain true when tested on synthetic and real dataset in a history matching exercise. Some practical aspects of binary map processes will be examined through the paper, as it is shown that it is more proper to use a derivative free optimisation algorithm and a proper metric should be more inclined to capture global features than local features. A performance comparison of metrics has been performed and applied to a geoscience real world problem.For binary seismic history matching, it is more appropriate to use a derivative free optimisation algorithm.In terms of similarly/dissimilarity measures, a "local" metric is a better choice for binary seismic history matching.When a binary filter is applied to an image, it is coarsening the search space.The combination of the Current Measure Metric and the binary mapping enhanced the final results.

We present and prove Polyakov-Alvarez type comparison formulas for the determinants of Friederichs extensions of Laplacians corresponding to conformally equivalent metrics on a compact Riemann surface with conical singularities. In particular, we find how the determinants depend on the orders of conical singularities. We also illustrate these general results with several examples: based on our Polyakov-Alvarez type formulas we recover known and obtain new explicit formulas for determinants of Laplacians on singular surfaces with and without boundary. In one of the examples we show that on the metrics of constant curvature on a sphere with two conical singularities and fixed area 4π the determinant of Friederichs Laplacian is unbounded from above and attains its local maximum on the metric of standard round sphere. In another example we deduce the famous Aurell-Salomon formula for the determinant of Friederichs Laplacian on polyhedra with spherical topology, thus providing the formula with mathematically rigorous proof.

In this work, the class of b-rectangular metric spaces is introduced, some fixed point results dealing with rational type contractions and almost generalized weakly contractive mappings are obtained. Certain examples are given to support the results.

We classify left invariant metrics on the 4-dimensional, simply connected, unimodular Lie groups up to automorphism. When the corresponding Lie algebra is of type (R), this is equivalent to classifying the left invariant metrics up to isometry, but in general the classification up to automorphism is finer than that up to isometry. In the abelian case, all left invariant metrics are isometric. In the nilpotent case, the space of metrics can have dimension 1 or 3. In the solvable case, the dimension can be 2, 4, or 5. There are two non-solvable 4-dimensional unimodular groups, and the space of metrics has dimension 6 in both of these cases.

An approximation modulus, the $\mathrm{AM}$-modulus, for a family of paths is introduced in a metric measure space ${(X,d,\nu)}$. It is shown that a function of bounded variation in X is a BV function on almost every path with respect to the $\mathrm{AM}$-modulus. Properties of the $\mathrm{AM}$-modulus, its relations to the usual ${M_{1}}$-modulus and to the BV-capacities of condensers are studied. Only minimal assumptions on X and the measure ν are employed.

In this paper we discuss the topological properties of quasi-partialb-metric spaces. The notion of quasi-partialb-metric space was introduced and fixed point theorem and coupled fixed point theorem on this space were studied. Here the concept of quasi-partialb-metric topology is discussed and notion of product of quasi-partialb-metric spaces is also introduced.

In this paper we study smooth metric measure spaces with boundary via the Bakry–Émery curvature and the weighted mean curvature of the boundary. We establish the weighted Laplacian comparison theorems and the upper bound estimates of the distance from any point of the manifold to its boundary. As applications, we derive lower bound estimates for the first Dirichlet eigenvalue.

Numerical analysis of data from international trade and ecological networks has shown that the non-linear fitness–complexity metric is the best candidate to rank nodes by importance in bipartite networks that exhibit a nested structure. Despite its relevance for real networks, the mathematical properties of the metric and its variants remain largely unexplored. Here, we perform an analytic and numeric study of the fitness–complexity metric and a new variant, called minimal extremal metric. We rigorously derive exact expressions for node scores for perfectly nested networks and show that these expressions explain the non-trivial convergence properties of the metrics. A comparison between the fitness–complexity metric and the minimal extremal metric on real data reveals that the latter can produce improved rankings if the input data are reliable.