Gromov-Monge quasi-metrics and distance distributions

Applications in data science, shape analysis and object classification frequently require maps between metric spaces which preserve geometry as faithfully as possible. In this paper, we combine the Monge formulation of optimal transport with the Gromov-Hausdorff distance construction to define a measure of the minimum amount of geometric distortion required to map one metric measure space onto another. We show that the resulting quantity, called Gromov-Monge distance, defines an extended quasi-metric on the space of isomorphism classes of metric measure spaces and that it can be promoted to a true metric on certain subclasses of mm-spaces. We also give precise comparisons between Gromov-Monge distance and several other metrics which have appeared previously, such as the Gromov-Wasserstein metric and the continuous Procrustes metric of Lipman, Al-Aifari and Daubechies. Finally, we derive polynomial-time computable lower bounds for Gromov-Monge distance. These lower bounds are expressed in terms of distance distributions, which are classical invariants of metric measure spaces summarizing the volume growth of metric balls. In the second half of the paper, which may be of independent interest, we study the discriminative power of these lower bounds for simple subclasses of metric measure spaces. We first consider the case of planar curves, where we give a counterexample to the Curve Histogram Conjecture of Brinkman and Olver. Our results on plane curves are then generalized to higher dimensional manifolds, where we prove some sphere characterization theorems for the distance distribution invariant. Finally, we consider several inverse problems on recovering a metric graph from a collection of localized versions of distance distributions. Results are derived by establishing connections with concepts from the fields of computational geometry and topological data analysis.Comment: Version 2: Added many new results and improved expositio

A Linear Transportation Lp Distance for Pattern Recognition

The transportation Lp distance, denoted TLp, has been proposed as a generalisation of Wasserstein Wp distances motivated by the property that it can be applied directly to colour or multi-channelled images, as well as multivariate time-series without normalisation or mass constraints. These distances, as with Wp, are powerful tools in modelling data with spatial or temporal perturbations. However, their computational cost can make them infeasible to apply to even moderate pattern recognition tasks. We propose linear versions of these distances and show that the linear TLp distance significantly improves over the linear Wp distance on signal processing tasks, whilst being several orders of magnitude faster to compute than the TLp distance

Optimization of Airfield Parking and Fuel Asset Dispersal to Maximize Survivability and Mission Capability Level

While the US focus for the majority of the past two decades has been on combatting insurgency and promoting stability in Southwest Asia, strategic focus is beginning to shift toward concerns of conflict with a near-peer state. Such conflict brings with it the risk of ballistic missile attack on air bases. With 26 conflicts worldwide in the past 100 years including attacks on air bases, new doctrine and modeling capacity are needed to enable the Department of Defense to continue use of vulnerable bases during conflict involving ballistic missiles. Several models have been developed to date for Air Force strategic planning use, but these models have limited use on a tactical level or for civil engineer use. This thesis presents the development of a novel model capable of identifying base layout characteristics for aprons and fuel depots to maximize dispersal and minimize impact on sortie generation times during normal operations. This model is implemented using multi-objective genetic algorithms to identify solutions that provide optimal tradeoffs between competing objectives and is assessed using an application example. These capabilities are expected to assist military engineers in the layout of parking plans and fuel depots that ensure maximum resilience while providing minimal impact to the user while enabling continued sortie generation in a contested region

Web-Based Dynamic Similarity Distance Tool

Similarity or distance measures is a well-known method and commonly used for calculating the distance between two samples of a dataset.Basically,the distance between the dataset samples is an important theory in multivariate analysis research.This paper proposes a tool that provides seven common distance methods that can be used in various research area.This tool is a web-based application which can be accessed through the internet browser.The objective of this tool is to introduce a web-based similarity distance application for many analysis and research purposes.Besides,a ranking method based on the Mean Average Precision is also implemented in this tool in order to increase the classification accuracies. This tool can process features that contain numerical values from any type of dataset

Big data clustering: Data preprocessing, variable selection, and dimension reduction

[no abstract available