# Spatio-Temporal Alignments: Optimal transport through space and time

**Authors:** Hicham Janati, Marco Cuturi, Alexandre Gramfort  
**arXiv:** [1910.03860](https://arxiv.org/abs/1910.03860)  
**Date:** 2019-10-09  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/1910.03860v3/

## Key points

- Authors address the challenging task of comparing data defined over both space and time
- Traditional methods like Dynamic Time Warping (DTW) overlook spatial shifts
- Proposed approach called Spatio-Temporal Alignments (STA) incorporates spatial differences using regularized optimal transport (OT)
- Temporal alignments are handled through a smooth variant of DTW known as soft-DTW
- Soft-DTW increases quadratically with time shifts
- Cost matrix within soft-DTW is computed using unbalanced OT for cases where observations are not normalized probabilities
- Experimental results on handwritten letters and brain imaging data validate the effectiveness of STA as a dissimilarity measure for spatio-temporal data
- STA offers a comprehensive solution for aligning spatio-temporal data by integrating spatial considerations into traditional alignment techniques

## Summary

In their paper titled "Spatio-Temporal Alignments: Optimal transport through space and time," authors Hicham Janati, Marco Cuturi, and Alexandre Gramfort address the challenging task of comparing data defined over both space and time. This task is complex due to the need to quantify spatial and temporal variability while also considering the chronological structure of the data. Traditional methods like Dynamic Time Warping (DTW) can compute optimal alignments between time series based on chronological order but often overlook spatial shifts. To overcome this limitation, the authors propose a novel approach called Spatio-Temporal Alignments (STA). STA introduces a differentiable formulation of DTW that incorporates spatial differences between time samples using regularized optimal transport (OT). Additionally, temporal alignments are handled through a smooth variant of DTW known as soft-DTW. The authors demonstrate a new property of soft-DTW, showing that it increases quadratically with time shifts. Furthermore, the cost matrix within soft-DTW is computed using unbalanced OT to accommodate cases where observations are not normalized probabilities. Experimental results on handwritten letters and brain imaging data validate the theoretical findings and showcase the effectiveness of STA as a dissimilarity measure for spatio-temporal data. Overall, this innovative approach offers a comprehensive solution for aligning spatio-temporal data by integrating spatial considerations into traditional alignment techniques. The research contributes valuable insights into handling complex datasets that evolve over both space and time, opening up new possibilities for analyzing and comparing such data in various fields.