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. Both TLP and WP assign a cost based on the transport distance (i.e. the "Lagrangian"model), the key difference between the distances is that TLP interprets the signal as a function whilst WP interprets the signal as a measure. Both distances 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. The linear Wasserstein distance was proposed as a method for projecting signals into a Euclidean space where the Euclidean distance is approximately the Wasserstein distance (more formally, this is a projection on to the tangent manifold). We propose linear versions of the TLP distance (LTLP) and we show significant improvement over the linear WP distance on signal processing tasks, whilst being several orders of magnitude faster to compute than the TLP distance.