# Dynamical Wasserstein Barycenters for Time-series Modeling

arXiv: https://arxiv.org/abs/2110.06741 · HTML: https://mlsanity.com/paper/2110.06741/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 38 references that are in the map.
- First posted on arXiv 13 Oct 2021.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [Structured Inference Networks for Nonlinear State Space Models](https://mlsanity.com/paper/1609.09869/) (2016)
- [Optimal transport for Gaussian mixture models](https://mlsanity.com/paper/1710.07876/) (2017)
- [A Brief Introduction to Manifold Optimization](https://mlsanity.com/paper/1906.05450/) (2019)
- [Optimal Transport Based Change Point Detection and Time Series Segment Clustering](https://mlsanity.com/paper/1911.01325/) (2019)
- [On Matched Filtering for Statistical Change Point Detection](https://mlsanity.com/paper/2006.05539/) (2020)
- [Manifold Optimization for Gaussian Mixture Models](https://mlsanity.com/paper/1506.07677/) (2015)

## Built on by (most cited first, 10 of 10)

- [Measure Estimation in the Barycentric Coding Model](https://mlsanity.com/paper/2201.12195/) (2022)
- [Nonparametric and Regularized Dynamical Wasserstein Barycenters for Sequential Observations](https://mlsanity.com/paper/2210.01918/) (2022)
- [On Rank Energy Statistics via Optimal Transport: Continuity, Convergence, and Change Point Detection](https://mlsanity.com/paper/2302.07964/) (2023)
- [Approximate Algorithms For k-Sparse Wasserstein Barycenter With Outliers](https://mlsanity.com/paper/2404.13401/) (2024)
- [Fast Gradient Computation for Gromov-Wasserstein Distance](https://mlsanity.com/paper/2404.08970/) (2024)
- [Dynamical Mode Recognition of Coupled Flame Oscillators by Supervised and Unsupervised Learning Approaches](https://mlsanity.com/paper/2404.17801/) (2024)
- [Dimensionality Reduction and Dynamical Mode Recognition of Circular Arrays of Flame Oscillators Using Deep Neural Network](https://mlsanity.com/paper/2312.02462/) (2023)
- [Linearized Wasserstein Barycenters: Synthesis, Analysis, Representational Capacity, and Applications](https://mlsanity.com/paper/2410.23602/) (2024)
- [A Novel Skip Orthogonal List for Dynamic Optimal Transport Problem](https://mlsanity.com/paper/2310.18446/) (2023)
- [On Barycenter Computation: Semi-Unbalanced Optimal Transport-based Method on Gaussians](https://mlsanity.com/paper/2410.08117/) (2024)

## Abstract (arXiv)

Many time series can be modeled as a sequence of segments representing high-level discrete states, such as running and walking in a human activity application. Flexible models should describe the system state and observations in stationary "pure-state" periods as well as transition periods between adjacent segments, such as a gradual slowdown between running and walking. However, most prior work assumes instantaneous transitions between pure discrete states. We propose a dynamical Wasserstein barycentric (DWB) model that estimates the system state over time as well as the data-generating distributions of pure states in an unsupervised manner. Our model assumes each pure state generates data from a multivariate normal distribution, and characterizes transitions between states via displacement-interpolation specified by the Wasserstein barycenter. The system state is represented by a barycentric weight vector which evolves over time via a random walk on the simplex. Parameter learning leverages the natural Riemannian geometry of Gaussian distributions under the Wasserstein distance, which leads to improved convergence speeds. Experiments on several human activity datasets show that our proposed DWB model accurately learns the generating distribution of pure states while improving state estimation for transition periods compared to the commonly used linear interpolation mixture models.