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.

Is there a novel Einstein–Gauss–Bonnet theory in four dimensions?

Abstract: No! We show that the field equations of Einstein–Gauss–Bonnet theory defined in generic $$D>4$$ dimensions split into two parts one of which always remains higher dimensional, and hence the theory does not have a non-trivial limit to $$D=4$$. Therefore, the recently introduced four-dimensional, novel, Einstein–Gauss–Bonnet theory does not admit an intrinsically four-dimensional definition, in terms of metric only, as such it does not exist in four dimensions. The solutions (the spacetime, the metric) always remain $$D>4$$ dimensional. As there is no canonical choice of 4 spacetime dimensions out of D dimensions for generic metrics, the theory is not well defined in four dimensions.

Central Moment Discrepancy (CMD) for Domain-Invariant Representation Learning

Abstract: The learning of domain-invariant representations in the context of domain adaptation with neural networks is considered. We propose a new regularization method that minimizes the discrepancy between domain-specific latent feature representations directly in the hidden activation space. Although some standard distribution matching approaches exist that can be interpreted as the matching of weighted sums of moments, e.g. Maximum Mean Discrepancy (MMD), an explicit order-wise matching of higher order moments has not been considered before. We propose to match the higher order central moments of probability distributions by means of order-wise moment differences. Our model does not require computationally expensive distance and kernel matrix computations. We utilize the equivalent representation of probability distributions by moment sequences to define a new distance function, called Central Moment Discrepancy (CMD). We prove that CMD is a metric on the set of probability distributions on a compact interval. We further prove that convergence of probability distributions on compact intervals w.r.t. the new metric implies convergence in distribution of the respective random variables. We test our approach on two different benchmark data sets for object recognition (Office) and sentiment analysis of product reviews (Amazon reviews). CMD achieves a new state-of-the-art performance on most domain adaptation tasks of Office and outperforms networks trained with MMD, Variational Fair Autoencoders and Domain Adversarial Neural Networks on Amazon reviews. In addition, a post-hoc parameter sensitivity analysis shows that the new approach is stable w.r.t. parameter changes in a certain interval. The source code of the experiments is publicly available.

Sapa: a multi-objective metric temporal planner

Abstract: Sapa is a domain-independent heuristic forward chaining planner that can handle durative actions, metric resource constraints, and deadline goals. It is designed to be capable of handling the multi-objective nature of metric temporal planning. Our technical contributions include (i) planning-graph based methods for deriving heuristics that are sensitive to both cost and makespan (ii) techniques for adjusting the heuristic estimates to take action interactions and metric resource limitations into account and (iii) a linear time greedy post-processing technique to improve execution flexibility of the solution plans. An implementation of Sapa using many of the techniques presented in this paper was one of the best domain independent planners for domains with metric and temporal constraints in the third International Planning Competition, held at AIPS-02. We describe the technical details of extracting the heuristics and present an empirical evaluation of the current implementation of Sapa.

A Round-trip Delay Metric for IPPM

Abstract: This memo defines a metric for round-trip delay of packets across Internet paths. [STANDARDS-TRACK]

A general limit theorem for recursive algorithms and combinatorial structures

Abstract: Limit laws are proven by the contraction method for random vectors of a recursive nature as they arise as parameters of combinatorial structures such as random trees or recursive algorithms, where we use the Zolotarev metric. In comparison to previous applications of this method, a general transfer theorem is derived which allows us to establish a limit law on the basis of the recursive structure and the asymptotics of the first and second moments of the sequence. In particular, a general asymptotic normality result is obtained by this theorem which typically cannot be handled by the more common $\ell_2$ metrics. As applications we derive quite automatically many asymptotic limit results ranging from the size of tries or $m$-ary search trees and path lengths in digital structures to mergesort and parameters of random recursive trees, which were previously shown by different methods one by one. We also obtain a related local density approximation result as well as a global approximation result. For the proofs of these results we establish that a smoothed density distance as well as a smoothed total variation distance can be estimated from above by the Zolotarev metric, which is the main tool in this article.