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 Lagrangianmodel), 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.

Automated summary

The transportation LP distance (TLP) is a generalization of the Wasserstein WP distance that can be applied directly to color or multi-channelled images, as well as multivariate time-series. TLP interprets signals as functions, while WP interprets signals as measures. Although both distances are powerful tools in modeling data with spatial or temporal perturbations, their computational cost can be prohibitively high for moderate pattern recognition tasks. The linear Wasserstein distance offers a method for projecting signals into a Euclidean space, and in this study, we propose linear versions of the TLP distance (LTLP) that show significant improvement over the linear WP distance in signal processing tasks while being several orders of magnitude faster to compute than the TLP distance.