Functional Inequalities for Metric-Preserving Functions with Respect to Intrinsic Metrics of Hyperbolic Type

We obtain functional inequalities for functions which are metric-preserving with respect to one of the following intrinsic metrics in a canonical plane domain: hyperbolic metric or some restrictions of the triangular ratio metric, respectively, of a Barrlund metric. The subadditivity turns out to be an essential property, being possessed by every function that is metric-preserving with respect to the hyperbolic metric and also by the composition with some specific function of every function that is metric-preserving with respect to some restriction of the triangular ratio metric or of a Barrlund metric. We partially answer an open question, proving that the hyperbolic arctangent is metric-preserving with respect to the restrictions of the triangular ratio metric on the unit disk to radial segments and to circles centered at origin.

Characterizations of canonically compactifiable graphs via intrinsic metrics and algebraic properties

We consider infinite graphs and the associated energy forms. We show that a graph is canonically compactifiable (i.e. all functions of finite energy are bounded) if and only if the underlying set is totally bounded with respect to any finite measure intrinsic metric. Furthermore, we show that a graph is canonically compactifiable if and only if the space of functions of finite energy is an algebra. These results answer questions in a recent work of Georgakopoulos, Haeseler, Keller, Lenz, and Wojciechowski.

CoNLL Dependency Parser: Extrinsic Evaluation through the Open Information Extraction task

Dependency Parsers (DP) are parsers that analyze dependencies between words in a sentence. Currently, dependency parser evaluation is a problem whose solutions are not well defined in the scientific community. Although the DP intrinsic metrics are the foremost choice of evaluation, extrinsic evaluation enables a different evaluation based on a downstream. Different results of DP can vary according to the domain task. Thus, this work applies an Open Information Extraction (OIE) method in Portuguese to provide an extrinsic evaluation of a set of CONLL Dependency Parsers. Our results demonstrate that there is a difference in the evaluation of Dependency Parsers considering a particular task.

Evaluating Natural Language Generation via Unbalanced Optimal Transport

Embedding-based evaluation measures have shown promising improvements on the correlation with human judgments in natural language generation. In these measures, various intrinsic metrics are used in the computation, including generalized precision, recall, F-score and the earth mover's distance. However, the relations between these metrics are unclear, making it difficult to determine which measure to use in real applications. In this paper, we provide an in-depth study on the relations between these metrics. Inspired by the optimal transportation theory, we prove that these metrics correspond to the optimal transport problem with different hard marginal constraints. However, these hard marginal constraints may cause the problem of incomplete and noisy matching in the evaluation process. Therefore we propose a family of new evaluation metrics, namely Lazy Earth Mover's Distances, based on the more general unbalanced optimal transport problem. Experimental results on WMT18 and WMT19 show that our proposed metrics have the ability to produce more consistent evaluation results with human judgements, as compared with existing intrinsic metrics.

Intrinsic Metric Formulas on Some Self-Similar Sets via the Code Representation

In recent years, intrinsic metrics have been described on various fractals with different formulas. The Sierpinski gasket is given as one of the fundamental models which defined the intrinsic metrics on them via the code representations of the points. In this paper, we obtain the explicit formulas of the intrinsic metrics on some self-similar sets (but not strictly self-similar), which are composed of different combinations of equilateral and right Sierpinski gaskets, respectively, by using the code representations of their points. We then express geometrical properties of these structures on their code sets and also give some illustrative examples.