# Measure Estimation in the Barycentric Coding Model

arXiv: https://arxiv.org/abs/2201.12195 · HTML: https://mlsanity.com/paper/2201.12195/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 55 references that are in the map.
- First posted on arXiv 28 Jan 2022.

## Builds on (most cited first, 18 of 18)

- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Faster Wasserstein Distance Estimation with the Sinkhorn Divergence](https://mlsanity.com/paper/2006.08172/) (2020)
- [Optimal Transport for Multi-source Domain Adaptation under Target Shift](https://mlsanity.com/paper/1803.04899/) (2018)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021)
- [Automatic Text Evaluation through the Lens of Wasserstein Barycenters](https://mlsanity.com/paper/2108.12463/) (2021)
- [Wasserstein Dictionary Learning: Optimal Transport-based unsupervised non-linear dictionary learning](https://mlsanity.com/paper/1708.01955/) (2017)
- [Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent](https://mlsanity.com/paper/2106.08502/) (2021)
- [Stochastic Wasserstein Barycenters](https://mlsanity.com/paper/1802.05757/) (2018)
- [Distilled Wasserstein Learning for Word Embedding and Topic Modeling](https://mlsanity.com/paper/1809.04705/) (2018)
- [Near-optimal estimation of smooth transport maps with kernel sums-of-squares](https://mlsanity.com/paper/2112.01907/) (2021)
- [Context Mover's Distance & Barycenters: Optimal Transport of Contexts for Building Representations](https://mlsanity.com/paper/1808.09663/) (2018)
- [Dynamical Wasserstein Barycenters for Time-series Modeling](https://mlsanity.com/paper/2110.06741/) (2021)
- [A Fast Globally Linearly Convergent Algorithm for the Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1809.04249/) (2018)

## Built on by (most cited first, 9 of 9)

- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [Acceptability Judgements via Examining the Topology of Attention Maps](https://mlsanity.com/paper/2205.09630/) (2022)
- [Linearized Wasserstein dimensionality reduction with approximation guarantees](https://mlsanity.com/paper/2302.07373/) (2023)
- [Tangential Wasserstein Projections](https://mlsanity.com/paper/2207.14727/) (2022)
- [Geometric Sparse Coding in Wasserstein Space](https://mlsanity.com/paper/2210.12135/) (2022)
- [A primer on optimal transport for causal inference with observational data](https://mlsanity.com/paper/2503.07811/) (2025)
- [Linearized Wasserstein Barycenters: Synthesis, Analysis, Representational Capacity, and Applications](https://mlsanity.com/paper/2410.23602/) (2024)
- [Static and Dynamic Approaches to Computing Barycenters of Probability Measures on Graphs](https://mlsanity.com/paper/2603.26940/) (2026)
- [Locality Regularized Reconstruction: Structured Sparsity and Delaunay Triangulations](https://mlsanity.com/paper/2405.00837/) (2024)

## Abstract (arXiv)

This paper considers the problem of measure estimation under the barycentric coding model (BCM), in which an unknown measure is assumed to belong to the set of Wasserstein-2 barycenters of a finite set of known measures. Estimating a measure under this model is equivalent to estimating the unknown barycentric coordinates. We provide novel geometrical, statistical, and computational insights for measure estimation under the BCM, consisting of three main results. Our first main result leverages the Riemannian geometry of Wasserstein-2 space to provide a procedure for recovering the barycentric coordinates as the solution to a quadratic optimization problem assuming access to the true reference measures. The essential geometric insight is that the parameters of this quadratic problem are determined by inner products between the optimal displacement maps from the given measure to the reference measures defining the BCM. Our second main result then establishes an algorithm for solving for the coordinates in the BCM when all the measures are observed empirically via i.i.d. samples. We prove precise rates of convergence for this algorithm -- determined by the smoothness of the underlying measures and their dimensionality -- thereby guaranteeing its statistical consistency. Finally, we demonstrate the utility of the BCM and associated estimation procedures in three application areas: (i) covariance estimation for Gaussian measures; (ii) image processing; and (iii) natural language processing.