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

TL;DR

This paper introduces a linear version of the transportation L^p distance, which is faster to compute and more effective for pattern recognition tasks involving images and time-series data.

Contribution

The paper proposes a linearized transportation L^p distance that enhances computational efficiency and improves performance over linear Wasserstein distances in pattern recognition.

Findings

Linear TL^p outperforms linear W^p in signal processing tasks.

Linear TL^p is several orders of magnitude faster than the original TL^p.

The method effectively handles multi-channel images and multivariate time-series.

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.