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Quadratically Regularized Optimal Transport: Existence and Multiplicity of Potentials

Marcel Nutz

TL;DR

This work analyzes quadratically regularized OT with general costs, proving the optimal coupling has density $d\pi_{*}/dP=(f+g-c)_{+}$ (with appropriate scaling) where $f,g$ are potentials. It establishes existence of these potentials in both discrete and continuous settings, details when potentials are non-unique (discrete) versus unique up to an additive constant (continuous), and characterizes the discrete multiplicity via connected components of the support. In the quadratic-cost continuous setting, the authors show a first sparsity guarantee: for small regularization, the optimal coupling concentrates near Brenier’s map graph, i.e., $\mathrm{spt}(\pi_{\varepsilon})$ remains near a sparse structure as $\varepsilon\to0$. The results rely on a Hilbert-space projection view of the primal problem, finite-dimensional approximations to construct potentials, and a careful limit argument to pass from approximations to a global dual representation. Overall, the paper unifies discrete and continuous analyses of potentials, clarifies when multiplicities arise, and provides a rigorous sparsity guarantee for quadratically regularized OT with practical implications for sparsity-driven applications.

Abstract

The optimal transport problem with quadratic regularization is useful when sparse couplings are desired. The density of the optimal coupling is described by two functions called potentials; equivalently, potentials can be defined as a solution of the dual problem. We prove the existence of potentials for a general square-integrable cost. Potentials are not necessarily unique, a phenomenon directly related to sparsity of the optimal support. For discrete problems, we describe the family of all potentials based on the connected components of the support, for a graph-theoretic notion of connectedness. On the other hand, we show that continuous problems have unique potentials under standard regularity assumptions, regardless of sparsity. Using potentials, we prove that the optimal support is indeed sparse for small regularization parameter in a continuous setting with quadratic cost, which seems to be the first theoretical guarantee for sparsity in this context.

Quadratically Regularized Optimal Transport: Existence and Multiplicity of Potentials

TL;DR

This work analyzes quadratically regularized OT with general costs, proving the optimal coupling has density (with appropriate scaling) where are potentials. It establishes existence of these potentials in both discrete and continuous settings, details when potentials are non-unique (discrete) versus unique up to an additive constant (continuous), and characterizes the discrete multiplicity via connected components of the support. In the quadratic-cost continuous setting, the authors show a first sparsity guarantee: for small regularization, the optimal coupling concentrates near Brenier’s map graph, i.e., remains near a sparse structure as . The results rely on a Hilbert-space projection view of the primal problem, finite-dimensional approximations to construct potentials, and a careful limit argument to pass from approximations to a global dual representation. Overall, the paper unifies discrete and continuous analyses of potentials, clarifies when multiplicities arise, and provides a rigorous sparsity guarantee for quadratically regularized OT with practical implications for sparsity-driven applications.

Abstract

The optimal transport problem with quadratic regularization is useful when sparse couplings are desired. The density of the optimal coupling is described by two functions called potentials; equivalently, potentials can be defined as a solution of the dual problem. We prove the existence of potentials for a general square-integrable cost. Potentials are not necessarily unique, a phenomenon directly related to sparsity of the optimal support. For discrete problems, we describe the family of all potentials based on the connected components of the support, for a graph-theoretic notion of connectedness. On the other hand, we show that continuous problems have unique potentials under standard regularity assumptions, regardless of sparsity. Using potentials, we prove that the optimal support is indeed sparse for small regularization parameter in a continuous setting with quadratic cost, which seems to be the first theoretical guarantee for sparsity in this context.
Paper Structure (15 sections, 20 theorems, 89 equations, 2 figures)

This paper contains 15 sections, 20 theorems, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Existence
  3. Multiplicity of Potentials
  4. Discrete Case
  5. Counterexample for Self-Transport
  6. Continuous Case
  7. Sparsity for Small Regularization
  8. Proof of \ref{['th:main']} (i)--(iii)
  9. Primal Existence
  10. Integrability Properties
  11. Weak Duality
  12. Construction of Potentials
  13. Self-Transport: Proof of \ref{['th:main']} (iv)
  14. Examples with Constant Cost
  15. Technical Remarks

Key Result

Theorem 2.2

Figures (2)

  • Figure 1: Illustration of a subset $E$ (colored area) of the square $\mathsf{X}\times\mathsf{Y}=[0,1]^{2}$ with three components (color coding).
  • Figure 2: Example for $\mathsf{X}=[0,1]$ with three components (the diagonal is displayed top-left to bottom-right, as for matrices). Here $\Lambda_{\rm low}$, $\Lambda_{\rm diag}$, $\Lambda_{\rm upp}$ each have one element.

Theorems & Definitions (53)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3: General $\varepsilon>0$
  • proof : Proof of (b)$\Leftrightarrow$(d)
  • Lemma 2.4
  • Lemma 2.5: Oscillation
  • proof
  • Lemma 2.6: Lipschitz potentials
  • proof
  • Example 3.1
  • ...and 43 more