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.

中文翻译：

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用于模式识别的线性运输 $\mathrm{L}^p$ 距离

运输 $\mathrm{L}^p$ 距离，表示为 $\mathrm{TL}^p$，已被提议作为 Wasserstein $\mathrm{W}^p$ 距离的推广，其动机是它可以是直接应用于彩色或多通道图像，以及没有归一化或质量约束的多元时间序列。这些距离与 $\mathrm{W}^p$ 一样，是对具有空间或时间扰动的数据进行建模的强大工具。然而，它们的计算成本可能使它们无法应用于即使是中等的模式识别任务。我们提出了这些距离的线性版本，并表明线性 $\mathrm{TL}^p$ 距离在信号处理任务上显着优于线性 $\mathrm{W}^p$ 距离，同时比线性 $\mathrm{W}^p$ 距离快几个数量级。计算比 $\mathrm{TL}^p$ 距离。