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

arXiv: https://arxiv.org/abs/1910.03860 · HTML: https://mlsanity.com/paper/1910.03860/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2019.

## Built on by (most cited first, 17 of 17)

- [Neural Unbalanced Optimal Transport via Cycle-Consistent Semi-Couplings](https://mlsanity.com/paper/2209.15621/) (2022)
- [A High-Quality Robust Diffusion Framework for Corrupted Dataset](https://mlsanity.com/paper/2311.17101/) (2023)
- [Unsupervised Ground Metric Learning using Wasserstein Singular Vectors](https://mlsanity.com/paper/2102.06278/) (2021)
- [Match-And-Deform: Time Series Domain Adaptation through Optimal Transport and Temporal Alignment](https://mlsanity.com/paper/2308.12686/) (2023)
- [Explanatory Model Monitoring to Understand the Effects of Feature Shifts on Performance](https://mlsanity.com/paper/2408.13648/) (2024)
- [Relative Translation Invariant Wasserstein Distance](https://mlsanity.com/paper/2409.02416/) (2024)
- [A Riemannian Approach to Ground Metric Learning for Optimal Transport](https://mlsanity.com/paper/2409.10085/) (2024)
- [Factored couplings in multi-marginal optimal transport via difference of convex programming](https://mlsanity.com/paper/2110.00629/) (2021)
- [Averaging Spatio-temporal Signals using Optimal Transport and Soft Alignments](https://mlsanity.com/paper/2203.05813/) (2022)
- [OTW: Optimal Transport Warping for Time Series](https://mlsanity.com/paper/2306.00620/) (2023)
- [PTLp: Partial Transport Lp Distances](https://mlsanity.com/paper/2307.13571/) (2023)
- [On the Convergence of Semi-Relaxed Sinkhorn with Marginal Constraint and OT Distance Gaps](https://mlsanity.com/paper/2205.13846/) (2022)
- [GeOT: A spatially explicit framework for evaluating spatio-temporal predictions](https://mlsanity.com/paper/2410.11709/) (2024)
- [An inexact Bregman proximal point method and its acceleration version for unbalanced optimal transport](https://mlsanity.com/paper/2402.16978/) (2024)
- [Uncertainty-DTW for Sequences and Visual Tokens](https://mlsanity.com/paper/2605.25110/) (2026)
- [Measuring Spatial Clustering via Metropolis-Hastings Diffusion Distance](https://mlsanity.com/paper/2607.14880/) (2026)
- [Accelerated Algorithm for Sparse Regularized Partial Optimal Transport](https://mlsanity.com/paper/2609.40075/) (2026)

## Abstract (arXiv)

Comparing data defined over space and time is notoriously hard, because it involves quantifying both spatial and temporal variability, while at the same time taking into account the chronological structure of data. Dynamic Time Warping (DTW) computes an optimal alignment between time series in agreement with the chronological order, but is inherently blind to spatial shifts. In this paper, we propose Spatio-Temporal Alignments (STA), a new differentiable formulation of DTW, in which spatial differences between time samples are accounted for using regularized optimal transport (OT). Our temporal alignments are handled through a smooth variant of DTW called soft-DTW, for which we prove a new property: soft-DTW increases quadratically with time shifts. The cost matrix within soft-DTW that we use are computed using unbalanced OT, to handle the case in which observations are not normalized probabilities. Experiments on handwritten letters and brain imaging data confirm our theoretical findings and illustrate the effectiveness of STA as a dissimilarity for spatio-temporal data.