Class Of Metric Spaces Research Articles

Intrinsic Lipschitz graphs in Carnot groups of step 2

We focus our attention on the notion of intrinsic Lipschitz graphs, inside a special class of metric spaces i.e. the Carnot groups. More precisely, we provide a characterization of locally intrinsic Lipschitz functions in Carnot groups of step 2 in terms of their intrinsic distributional gradients.

Multi-marginal entropy-transport with repulsive cost

In this paper we study theoretical properties of the entropy-transport functional with repulsive cost functions. We provide sufficient conditions for the existence of a minimizer in a class of metric spaces and prove the Gamma -convergence of the entropy-transport functional to a multi-marginal optimal transport problem with a repulsive cost. We point out that our construction can deal with the case when the space X is a domain in {mathbb {R}}^d, answering a question raised in Benamou et al. (Numer Math 142:33–54, 2019). Finally, we also prove the entropy-regularized version of the Kantorovich duality.

Re-ranking via local embeddings: A use case with permutation-based indexing and the nSimplex projection

Approximate Nearest Neighbor (ANN) search is a prevalent paradigm for searching intrinsically high dimensional objects in large-scale data sets. Recently, the permutation-based approach for ANN has attracted a lot of interest due to its versatility in being used in the more general class of metric spaces. In this approach, the entire database is ranked by a permutation distance to the query. Typically, permutations allow the efficient selection of a candidate set of results, but typically to achieve high recall or precision this set has to be reviewed using the original metric and data. This can lead to a sizeable percentage of the database being recalled, along with many expensive distance calculations.To reduce the number of metric computations and the number of database elements accessed, we propose here a re-ranking based on a local embedding using the nSimplex projection. The nSimplex projection produces Euclidean vectors from objects in metric spaces which possess the n-point property. The mapping is obtained from the distances to a set of reference objects, and the original metric can be lower bounded and upper bounded by the Euclidean distance of objects sharing the same set of references.Our approach is particularly advantageous for extensive databases or expensive metric function. We reuse the distances computed in the permutations in the first stage, and hence the memory footprint of the index is not increased.An extensive experimental evaluation of our approach is presented, demonstrating excellent results even on a set of hundreds of millions of objects.

Fréchet change-point detection

We propose a method to infer the presence and location of change-points in the distribution of a sequence of independent data taking values in a general metric space, where change-points are viewed as locations at which the distribution of the data sequence changes abruptly in terms of either its Fréchet mean, Fréchet variance or both. The proposed method is based on comparisons of Fréchet variances before and after putative change-point locations and does not require a tuning parameter, except for the specification of cut-off intervals near the endpoints where change-points are assumed not to occur. Our results include theoretical guarantees for consistency of the test under contiguous alternatives when a change-point exists and also for consistency of the estimated location of the change-point, if it exists, where, under the null hypothesis of no change-point, the limit distribution of the proposed scan function is the square of a standardized Brownian bridge. These consistency results are applicable for a broad class of metric spaces under mild entropy conditions. Examples include the space of univariate probability distributions and the space of graph Laplacians for networks. Simulation studies demonstrate the effectiveness of the proposed methods, both for inferring the presence of a change-point and estimating its location. We also develop theory that justifies bootstrap-based inference and illustrate the new approach with sequences of maternal fertility distributions and communication networks.

Self-Contracted Curves in Spaces With Weak Lower Curvature Bound

Abstract We show that bounded self-contracted curves are rectifiable in metric spaces with weak lower curvature bound in a sense we introduce in this article. This class of spaces is wide and includes, for example, finite-dimensional Alexandrov spaces of curvature bounded below and Berwald spaces of nonnegative flag curvature. (To be more precise, our condition is regarded as a strengthened doubling condition and holds also for a certain class of metric spaces with upper curvature bound.) We also provide the non-embeddability of large snowflakes into (balls in) metric spaces in the same class. We follow the strategy of the last author’s previous paper based on the small rough angle condition, where spaces with upper curvature bound are considered. Here we apply this strategy to spaces with lower curvature bound.

On the Gromov-Hausdorff limit of metric spaces

Abstract In this paper, we show that a class of metric spaces determined by a continuous function f, which defines on the metric space of all real, n × n-matrices m is closed under the Gromov-Hausdorff convergence. This conclusion can be used to prove some metric properties of metric space is stable under the Gromov-Hausdorff convergence. Secondly, we consider the stability problem in Gromov hyperbolic space and show that if a sequence of Gromov hyperbolic spaces (Xn , dn ) is said to converge to (X, d) in the sense of Gromov-Hausdorff convergence, then the Gromov hyperbolicity δ(Xn ) of (Xn , dn ) tends to the Gromov hyperbolicity δ(X) of (X, d).

Symmetric Strong Diameter Two Property

We study Banach spaces with the property that, given a finite number of slices of the unit ball, there exists a direction such that all these slices contain a line segment of length almost 2 in this direction. This property was recently named the symmetric strong diameter two property by Abrahamsen, Nygaard, and Poldvere. The symmetric strong diameter two property is not just formally stronger than the strong diameter two property (finite convex combinations of slices have diameter 2). We show that the symmetric strong diameter two property is only preserved by $$\ell _\infty $$ -sums, and working with weak star slices we show that $$\text {Lip}_0(M)$$ have the weak star version of the property for several classes of metric spaces M.

FINITE FLAT SPACES

We say that a finite metric space $X$ can be embedded almost isometrically into a class of metric spaces $C$, if for every $\epsilon > 0$ there exists an embedding of $X$ into one of the elements of $C$ with the bi-Lipschitz distortion less then $1 + \epsilon$. We show that almost isometric embeddability conditions are equal for following classes of spaces (a) Quotients of Euclidean spaces by isometric actions of finite groups, (b) $L_2$-Wasserstein spaces over Euclidean spaces, (c) Compact flat manifolds, (d) Compact flat orbifolds, (e) Quotients of connected compact bi-invariant Lie groups by isometric actions of compact Lie groups. (This one is the most surprising.) We call spaces which satisfy this conditions finite flat spaces. The question about synthetic definition naturally arises. Since Markov type constants depend only on finite subsets we can conclude that connected compact bi-invariant Lie groups and their quotients have Markov type $2$ with constant $1$.

On Interpolative Hardy-Rogers Type Contractions

By using an interpolative approach, we recognize the Hardy-Rogers fixed point theorem in the class of metric spaces. The obtained result is supported by some examples. We also give the partial metric case, according to our result.

Indicators of Similarity and Dissimilarity of Multi-Attribute Objects in the Metric Spaces of Sets and Multisets

The concepts of the similarity and dissimilarity (difference) of analyzed objects play an important role in many theoretical and practical problems of decision making, artificial intelligence, pattern recognition, processing of heterogeneous information, etc. The similarity or dissimilarity of objects is usually estimated by their proximity in the attribute space. This paper considers new classes of metric spaces of finite, bounded, and measurable sets and multisets. The possibilities are shown for using new types of metrics to evaluate the similarity or dissimilarity of multi-attribute objects that are present in several instances with differing values of attributes and are represented by multisets of attributes.

Effective Hausdorff Dimension in General Metric Spaces

We introduce the concept of effective dimension for a wide class of metric spaces whose metric is not necessarily based on a measure. Effective dimension was defined by Lutz (Inf. Comput., 187(1), 49–79, 2003) for Cantor space and has also been extended to Euclidean space. Lutz effectivization uses gambling, in particular the concept of gale and supergale, our extension of Hausdorff dimension to other metric spaces is also based on a supergale characterization of dimension, which in practice avoids an extra quantifier present in the classical definition of dimension that is based on Hausdorff measure and therefore allows effectivization for small time-bounds. We present here the concept of constructive dimension and its characterization in terms of Kolmogorov complexity, for which we extend the concept of Kolmogorov complexity to any metric space defining the Kolmogorov complexity of a point at a certain precision. Further research directions are indicated.

Locality of the Heat Kernel on Metric Measure Spaces

We will discuss what it means for a general heat kernel on a metric measure space to be local. We show that the Wiener measure associated to Brownian motion is local. Next we show that locality of the Wiener measure plus a suitable decay bound of the heat kernel implies locality of the heat kernel. We define a class of metric spaces we call manifold-like that satisfy the prerequisites for these theorems. This class includes Riemannian manifolds, metric graphs, products and some quotients of these as well as a number of more singular spaces. There exists a natural Dirichlet form based on the Laplacian on manifold-like spaces and we show that the associated Wiener measure and heat kernel are both local. These results unify and generalise facts known for manifolds and metric graphs. They provide a useful tool for computing heat kernel asymptotics for a large class of metric spaces. As an application we compute the heat kernel asymptotics for two identical particles living on a metric graph.

Tropical convexity in Riesz spaces

Various properties of tropical convex sets of the positive cone of a Riesz space are established. Under suitable, but standard assumptions, we show that they have fixed points properties, for single valued and multivalued maps, analogous to those associated to the usual convex sets in topological vector spaces. Selection theorems and extension theorems similar to those of Michael and Dugundji also hold and, in the appropriate setting, tropical convex sets are absolute extensors for the class of metric spaces and absolute retracts. We also give some applications to standard problems of mathematical economics: Nash equilibria, existence of elements for non transitive preference relations and existence of equilibria for abstract economies in Riesz spaces under tropical convexity assumptions. This work can be conceived as a contribution to infinite dimensional idempotent analysis.

Minimal universal metric spaces

Let M be a class of metric spaces. A metric space Y is minimal M-universal if every X 2 M can be isometrically embedded in Y; but there are no proper subsets of Y; which satisfy this property. We find sucient conditions under which M has a minimal M-universal metric space and sucient conditions under which M has no minimal universal metric spaces. We generalize the notion of minimal M-universal metric space to notion of minimal M-universal class of metric spaces and prove the uniqueness, up to an isomorphism, for these classes. Examples of minimal universal metric spaces are constructed for some subclasses of the class of metric spaces X; which possesses the following property. Among every three distinct points of X there is one point lying between the other two points.

Large-scale rank of Teichmüller space

Let X be quasi-isometric to either the mapping class group equipped with the word metric, or to Teichmuller space equipped with either the Teichmuller metric or the Weil-Petersson metric. We introduce a unified approach to study the coarse geometry of these spaces. We show that the quasi-Lipschitz image in X of a box in R^n is locally near a standard model of a flat in X. As a consequence, we show that, for all these spaces, the geometric rank and the topological rank are equal. The methods are axiomatic and apply to a larger class of metric spaces.

Geometries Induced by Logarithmic Oscillations as Examples of Gromov Hyperbolic Spaces

We explore the connection between the geometries generated by logarithmic oscillations and the class of metric spaces satisfying the Gromov hyperbolicity condition. We investigate the most fundamental examples, inspired from classical geometries, e.g. the Euclidean distance on the infinite strip or Hilbert’s distance on the unit disk. We continue our study with the Barbilian’s distance, which historically appeared as a natural extension of a model of hyperbolic geometry. We introduce and investigate a new metric, called the stabilizing metric. In a natural development, we explore a class of extensions of this distance which, under some analytic conditions, produces infinitely many new examples of Gromov hyperbolic metric spaces. Using similar procedures, we construct Vuorinen’s stabilizing metric and its extensions, and we discuss their Gromov hyperbolicity.

Fixed point theorems for metric spaces with a conical geodesic bicombing

We derive two fixed point theorems for a class of metric spaces that includes all Banach spaces and all complete Busemann spaces. We obtain our results by the use of a $1$-Lipschitz barycenter construction and an existence result for invariant Radon probability measures. Furthermore, we construct a bounded complete Busemann space that admits an isometry without fixed points.

MapReduce and streaming algorithms for diversity maximization in metric spaces of bounded doubling dimension

Given a dataset of points in a metric space and an integer k , a diversity maximization problem requires determining a subset of k points maximizing some diversity objective measure, e.g., the minimum or the average distance between two points in the subset. Diversity maximization is computationally hard, hence only approximate solutions can be hoped for. Although its applications are mainly in massive data analysis, most of the past research on diversity maximization focused on the sequential setting. In this work we present space and pass/round-efficient diversity maximization algorithms for the Streaming and MapReduce models and analyze their approximation guarantees for the relevant class of metric spaces of bounded doubling dimension. Like other approaches in the literature, our algorithms rely on the determination of high-quality core-sets, i.e., (much) smaller subsets of the input which contain good approximations to the optimal solution for the whole input. For a variety of diversity objective functions, our algorithms attain an ( α + ε )-approximation ratio, for any constant ε > 0, where α is the best approximation ratio achieved by a polynomial-time, linear-space sequential algorithm for the same diversity objective. This improves substantially over the approximation ratios attainable in Streaming and MapReduce by state-of-the-art algorithms for general metric spaces. We provide extensive experimental evidence of the effectiveness of our algorithms on both real world and synthetic datasets, scaling up to over a billion points.

More on variants of complete metric spaces

Some classes of metric spaces satisfying properties stronger than completeness but weaker than compactness have been studied by many authors over the years. One such significant family consists of those metric spaces on which every real-valued continuous function is uniformly continuous, which are widely known as Atsuji spaces or UC spaces. Recently in 2014, two new kinds of complete metric spaces are introduced, namely Bourbaki-complete and cofinally Bourbaki-complete metric spaces, whose idea has come from some new classes of sequences acting as generalizations of Cauchy sequences. Our major goal is to give several new equivalent conditions for metric spaces whose completions are one of the aforesaid spaces, especially in terms of some functions, sequences and geometric functionals.

Two classes of metric spaces

<p>The class of metric spaces (X,d) known as small-determined spaces, introduced by Garrido and Jaramillo, are properly defined by means of some type of real-valued Lipschitz functions on X. On the other hand, B-simple metric spaces introduced by Hejcman are defined in terms of some kind of bornologies of bounded subsets of X. In this note we present a common framework where both classes of metric spaces can be studied which allows us to see not only the relationships between them but also to obtain new internal characterizations of these metric properties.</p>