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Linearized Wasserstein Barycenters: Synthesis, Analysis, Representational Capacity, and Applications

Matthew Werenski, Brendan Mallery, Shuchin Aeron, James M. Murphy

TL;DR

A closed-form solution to the variational problem characterizing the probability measures in the LBCM is provided and equivalence of the LBCM to the set of 2-Wasserstein barycenters in the special case of compatible measures is established.

Abstract

We propose the linear barycentric coding model (LBCM) which utilizes the linear optimal transport (LOT) metric for analysis and synthesis of probability measures. We provide a closed-form solution to the variational problem characterizing the probability measures in the LBCM and establish equivalence of the LBCM to the set of 2-Wasserstein barycenters in the special case of compatible measures. Computational methods for synthesizing and analyzing measures in the LBCM are developed with finite sample guarantees. One of our main theoretical contributions is to identify an LBCM, expressed in terms of a simple family, which is sufficient to express all probability measures on the closed unit interval. We show that a natural analogous construction of an LBCM in 2 dimensions fails, and we leave it as an open problem to identify the proper extension in more than 1 dimension. We conclude by demonstrating the utility of LBCM for covariance estimation and data imputation.

Linearized Wasserstein Barycenters: Synthesis, Analysis, Representational Capacity, and Applications

TL;DR

A closed-form solution to the variational problem characterizing the probability measures in the LBCM is provided and equivalence of the LBCM to the set of 2-Wasserstein barycenters in the special case of compatible measures is established.

Abstract

We propose the linear barycentric coding model (LBCM) which utilizes the linear optimal transport (LOT) metric for analysis and synthesis of probability measures. We provide a closed-form solution to the variational problem characterizing the probability measures in the LBCM and establish equivalence of the LBCM to the set of 2-Wasserstein barycenters in the special case of compatible measures. Computational methods for synthesizing and analyzing measures in the LBCM are developed with finite sample guarantees. One of our main theoretical contributions is to identify an LBCM, expressed in terms of a simple family, which is sufficient to express all probability measures on the closed unit interval. We show that a natural analogous construction of an LBCM in 2 dimensions fails, and we leave it as an open problem to identify the proper extension in more than 1 dimension. We conclude by demonstrating the utility of LBCM for covariance estimation and data imputation.

Paper Structure

This paper contains 32 sections, 23 theorems, 117 equations, 6 figures, 2 tables, 9 algorithms.

Table of Contents

  1. Introduction
  2. Summary of Contributions
  3. Background
  4. Related Work
  5. Properties of the LBCM
  6. Synthesis and Analysis in the LBCM
  7. Plug-in Estimation for the Synthesis and Analysis
  8. Representation Via LBCM
  9. Density of the LBCM in $\mathcal{P}([a,b])$
  10. Capacity in Higher Dimensions
  11. Applications
  12. Covariance Estimation
  13. MNIST Reconstruction
  14. Conclusion
  15. Proofs from Section \ref{['sec:BasicProperties']}
  16. ...and 17 more sections

Key Result

Theorem 1

Let $\mu \in \mathcal{P}_{2,ac}(\mathbb{R}^{d})$ and $\nu \in \mathcal{P}_{2}(\mathbb{R}^{d})$. Then there exists a unique (up to $\mu$-a.e. equivalence) map $T^{*}$ such that $T^{*} \# \mu = \nu$, and Moreover, there exists a convex function $\varphi$ such that $\nabla\varphi=T^{*}$.

Figures (6)

  • Figure 1: Recovery of the covariance matrix and coordinate in logarithmic scale.
  • Figure 2: Reconstruction of occluded digits using the $W_{2}$BCM, LBCM, and a linear method. The base measure in Column 4 is the Double Checker in Figure \ref{['fig : base measures']}.
  • Figure 3: Structure of $C_0$ and the sets $R_1,R_2,R_3$. Each set contains its boundary (shown in blue) and the intersection point is determined by $b$.
  • Figure 4: Counter example in Theorem \ref{['thm : questions negative']}. On the left we have a typical element of $\mathcal{V}(C_0)$, in the middle the gradient of $\phi_0$, and on the right a numerically obtained optimal reconstruction of $\phi_0$ using a convex combination of elements in $\mathcal{V}(C_0)$ represented as vector fields. Arrow color corresponds to the magnitude of the first coordinate.
  • Figure 5: Base measures used in Figure \ref{['fig : lbcm reconstructions']}
  • ...and 1 more figures

Theorems & Definitions (50)

  • Definition 1
  • Definition 2
  • Theorem 1: Brenier
  • Definition 3
  • Proposition 1
  • Definition 4
  • Proposition 2
  • Remark 1
  • Theorem 2: pooladian2021entropic, Theorem 3
  • Remark 2
  • ...and 40 more