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Knothe-Rosenblatt maps via soft-constrained optimal transport

Ricardo Baptista, Franca Hoffmann, Minh Van Hoang Nguyen, Benjamin Zhang

Abstract

In the theory of optimal transport, the Knothe-Rosenblatt (KR) rearrangement provides an explicit construction to map between two probability measures by building one-dimensional transformations from the marginal conditionals of one measure to the other. The KR map has shown to be useful in different realms of mathematics and statistics, from proving functional inequalities to designing methodologies for sampling conditional distributions. It is known that the KR rearrangement can be obtained as the limit of a sequence of optimal transport maps with a weighted quadratic cost. We extend these results in this work by showing that one can obtain the KR map as a limit of maps that solve a relaxation of the weighted-cost optimal transport problem with a soft-constraint for the target distribution. In addition, we show that this procedure also applies to the construction of triangular velocity fields via dynamic optimal transport yielding optimal velocity fields. This justifies various variational methodologies for estimating KR maps in practice by minimizing a divergence between the target and pushforward measure through an approximate map. Moreover, it opens the possibilities for novel static and dynamic OT estimators for KR maps.

Knothe-Rosenblatt maps via soft-constrained optimal transport

Abstract

In the theory of optimal transport, the Knothe-Rosenblatt (KR) rearrangement provides an explicit construction to map between two probability measures by building one-dimensional transformations from the marginal conditionals of one measure to the other. The KR map has shown to be useful in different realms of mathematics and statistics, from proving functional inequalities to designing methodologies for sampling conditional distributions. It is known that the KR rearrangement can be obtained as the limit of a sequence of optimal transport maps with a weighted quadratic cost. We extend these results in this work by showing that one can obtain the KR map as a limit of maps that solve a relaxation of the weighted-cost optimal transport problem with a soft-constraint for the target distribution. In addition, we show that this procedure also applies to the construction of triangular velocity fields via dynamic optimal transport yielding optimal velocity fields. This justifies various variational methodologies for estimating KR maps in practice by minimizing a divergence between the target and pushforward measure through an approximate map. Moreover, it opens the possibilities for novel static and dynamic OT estimators for KR maps.

Paper Structure

This paper contains 19 sections, 20 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Organization
  3. Background, Notation and Assumptions
  4. Formulations of Optimal Transport
  5. Triangular Maps and Assumptions.
  6. Notation
  7. Knothe-Rosenblatt Maps via Soft-constraints
  8. Statement of Main Results
  9. Convergence to a soft-constrained Knothe-Rosenblatt map
  10. Convergence to the hard-constraint Knothe-Rosenblatt map
  11. Knothe-Rosenblatt Velocity Fields
  12. Discussion
  13. Acknowledgments
  14. Proof of \ref{['thm:kr stab']}
  15. Equivalence of the $(x_1,y_1)$ marginals
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $c(x, y)=\|x-y\|^2$ (or equivalently $c_\epsilon(x, y)=\|x-y\|_\epsilon^2)$. Suppose that $\mu$ and $\nu$ have finite second moment, and that $\mu \ll d x$. Then there exists a unique optimal plan $\gamma$. In addition, $\gamma=(id \times T)_{\sharp} \mu$ and $T=\nabla \varphi$ for some convex f

Figures (1)

  • Figure 1: "Commutative diagram" of limits between the minimizers of various optimal transport problems we consider in this work.

Theorems & Definitions (42)

  • Theorem 2.1: Brenier's theorem Brenier1991PolarFA
  • Theorem 2.2: Theorem 2.1 in carlier2008knothes
  • Definition 3.1: Soft-constrained Kantorovich problem
  • Definition 3.2: Hard-constrained Kantorovich Problem
  • Remark 3.3
  • Theorem 3.4: From OT to KR
  • Remark 3.5
  • proof : Proof Sketch
  • Theorem 3.6: Convergence to a perturbed Knothe-Rosenblatt map
  • proof
  • ...and 32 more