  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.
