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.