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Convex relaxation approaches for high-dimensional optimal transport

Yuehaw Khoo, Tianyun Tang

TL;DR

This work addresses the computational and statistical challenges of high-dimensional optimal transport by introducing convex relaxation frameworks that approximate the OT coupling using only low-order marginals and cluster moments. The marginal relaxation and the cluster moment relaxation yield tractable semidefinite programs that provide computable lower bounds on the OT cost and scale favorably with dimension, especially under correlative sparsity in Gaussian settings. Theoretical analysis shows exactness for complete sparsity and exponential convergence for sparse graphs, and the authors also describe how to extract transport maps from the relaxations for use in generative modeling. Numerical experiments across Gaussian, Ising, and Ginzburg–Landau models demonstrate improved efficiency and accuracy, and show that transport maps derived from these relaxations can outperform neural-network-based approaches in some regimes. Overall, the paper presents convex relaxation as a viable dimension-reduction strategy for scaling OT to high-dimensional problems with practical implications for learning and generative modeling.

Abstract

Optimal transport (OT) is a powerful tool in mathematics and data science but faces severe computational and statistical challenges in high dimensions. We propose convex relaxation approaches based on marginal and cluster moment relaxations that exploit locality and correlative sparsity in the distributions. These methods approximate high-dimensional couplings using low-order marginals and sparse moment statistics, yielding semidefinite programs that provide lower bounds on the OT cost with greatly reduced complexity. For Gaussian distributions with sparse correlations, we prove reductions in both computational and sample complexity, and experiments show the approach also works well for non-Gaussian cases. In addition, we demonstrate how to extract transport maps from our relaxations, offering a simpler and interpretable alternative to neural networks in generative modeling. Our results suggest that convex relaxations can provide a promising path for dimension reduction in high-dimensional OT.

Convex relaxation approaches for high-dimensional optimal transport

TL;DR

This work addresses the computational and statistical challenges of high-dimensional optimal transport by introducing convex relaxation frameworks that approximate the OT coupling using only low-order marginals and cluster moments. The marginal relaxation and the cluster moment relaxation yield tractable semidefinite programs that provide computable lower bounds on the OT cost and scale favorably with dimension, especially under correlative sparsity in Gaussian settings. Theoretical analysis shows exactness for complete sparsity and exponential convergence for sparse graphs, and the authors also describe how to extract transport maps from the relaxations for use in generative modeling. Numerical experiments across Gaussian, Ising, and Ginzburg–Landau models demonstrate improved efficiency and accuracy, and show that transport maps derived from these relaxations can outperform neural-network-based approaches in some regimes. Overall, the paper presents convex relaxation as a viable dimension-reduction strategy for scaling OT to high-dimensional problems with practical implications for learning and generative modeling.

Abstract

Optimal transport (OT) is a powerful tool in mathematics and data science but faces severe computational and statistical challenges in high dimensions. We propose convex relaxation approaches based on marginal and cluster moment relaxations that exploit locality and correlative sparsity in the distributions. These methods approximate high-dimensional couplings using low-order marginals and sparse moment statistics, yielding semidefinite programs that provide lower bounds on the OT cost with greatly reduced complexity. For Gaussian distributions with sparse correlations, we prove reductions in both computational and sample complexity, and experiments show the approach also works well for non-Gaussian cases. In addition, we demonstrate how to extract transport maps from our relaxations, offering a simpler and interpretable alternative to neural networks in generative modeling. Our results suggest that convex relaxations can provide a promising path for dimension reduction in high-dimensional OT.

Paper Structure

This paper contains 17 sections, 4 theorems, 63 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Optimal transport
  3. Curse of dimensionality
  4. Our contributions
  5. Organization
  6. Convex relaxations
  7. Preliminaries
  8. Marginal relaxation
  9. Cluster moment relaxation
  10. Theoretical analysis of Gaussian distribution
  11. Extracting a transport map.
  12. Numerical Experiments
  13. Gaussian distribution
  14. Ising Model
  15. Ginzburg--Landau Model
  16. ...and 2 more sections

Key Result

Lemma 3.1

Consider the linear SDP where each $A_i\in \mathbb{S}(G,0)$ for $i\in[m]\sqcup\{0\}$ (Definition itm:third). If $G$ is chordal with maximal cliques $V_1,\ldots,V_p\subset[n]$, then lSDP is equivalent to which is a multi-block SDP with block sizes $|V_1|,|V_2|,\ldots,|V_p|$.

Figures (7)

  • Figure 1: Step 1. Partition $x$ and $y$ into $K$ clusters $(x_1;x_2;\cdots;x_K)$ and $(y_1;y_2;\cdots;y_K)$. Construct a local coupling within each cluster: $z_k=(x_k,y_k)\sim\pi_k$. The mean-field approximation $\otimes_{k=1}^K \pi_k$ of $\pi$ is exact if the $z_k$'s are independent.
  • Figure 2: Step 2. Add pairwise couplings between correlated clusters: $(z_i,z_j)\sim \pi_{ij},$ consistent with marginals ${\rm P}_{z_i}(\pi_{ij})=\pi_i,{\rm P}_{z_j}(\pi_{ij})=\pi_j$ and satisfying the PSD constraint \ref{['PSDcons']}. In this example, the reference graph $\mathcal{G}$ (Definition \ref{['itm:third']}) is a triangle.
  • Figure 3: Cluster moment relaxation \ref{['GSmom']} for OT between Gaussian distributions. The correlation sparsity pattern $G$ of Gaussian distributions is a path. The reference graph $\mathcal{G}$ is chosen as $G^h$ for various correlation radius $h$ (Definition \ref{['itm:third']}).
  • Figure 4: Comparison of cluster moment relaxation \ref{['OTmom-2']} and the vanilla OT solver for \ref{['OT']} between Gaussian distributions. The reference graph for \ref{['OTmom-2']} is $\mathcal{G}=G^h$ (Definition \ref{['itm:third']}), such that $G$ is a path graph and the correlation radius $h=5.$
  • Figure 5: Comparison of marginal relaxation and traditional OT solver for \ref{['OT']} between Ising models with $(J_1,h_1,\beta_1)=(J_2,h_2,\beta_2)=(-1,0.2,0.3).$$\omega=1$ is the size of clusters in marginal relaxation defined in \ref{['defiomegx']} and \ref{['defiomegy']}. The reference graph $\mathcal{G}$ (Definition \ref{['itm:third']}) is a path graph.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • Lemma A.1
  • proof
  • ...and 1 more