A Linear Transportation ${\mathrm{L}}^p$ Distance for Pattern Recognition

Abstract

The transportation ${\mathrm{L}}^p$ distance, denoted ${\mathrm{TL}}^p$, has been proposed as a generalisation of Wasserstein ${\mathrm{W}}^p$ 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 ${\mathrm{W}}^p$, 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 ${\mathrm{TL}}^p$ distance significantly improves over the linear ${\mathrm{W}}^p$ distance on signal processing tasks, whilst being several orders of magnitude faster to compute than the ${\mathrm{TL}}^p$ distance.

References in corpus (8)

• Natural Scales in Geographical Patterns
• Fast Computation of Wasserstein Barycenters
• Learning with a Wasserstein Loss
• Bayesian Inference with Optimal Maps
• Continuum limit of total variation on point clouds
• An efficient linear programming method for Optimal Transportation
• Linearized Optimal Transport for Collider Events
• Multiscale Strategies for Computing Optimal Transport