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Convergence rate of entropy-regularized multi-marginal optimal transport costs

Luca Nenna, Paul Pegon

TL;DR

This work analyzes how entropic regularization affects multi-marginal OT costs as the regularization parameter $\varepsilon$ vanishes. It extends prior two-marginal results to the multi-marginal setting, employing a multi-marginal block-approximation and Alexandrov-type arguments to obtain dimension-dependent upper bounds and a signature-based lower bound tied to the mixed second derivatives of the cost. The key contributions include an upper bound $MOT_\varepsilon \le MOT_0 + (\sum_i d_i - \max_i d_i) \varepsilon \log(1/\varepsilon) + O(\varepsilon)$ for locally Lipschitz costs, a refined bound with a $1/2$ factor for locally semi-concave costs, and a lower bound $MOT_\varepsilon \ge MOT_0 + (\kappa/2) \varepsilon \log(1/\varepsilon) - C_* \varepsilon$ under a signature condition; in several cases the bounds match, revealing cases where the optimal plan is deterministic. These results clarify how the geometry of marginals and cost curvature govern the convergence rate, with implications for numerical schemes solving entropic MOT problems. Overall, the paper advances understanding of entropy-regularized MOT by linking convergence rates to marginal dimension and Hessian structure rather than the regularized plan itself.

Abstract

We investigate the convergence rate of multi-marginal optimal transport costs that are regularized with the Boltzmann-Shannon entropy, as the noise parameter $\varepsilon$ tends to $0$. We establish lower and upper bounds on the difference with the unregularized cost of the form $C\varepsilon\log(1/\varepsilon)+O(\varepsilon)$ for some explicit dimensional constants $C$ depending on the marginals and on the ground cost, but not on the optimal transport plans themselves. Upper bounds are obtained for Lipschitz costs or locally semi-concave costs for a finer estimate, and lower bounds for $\mathscr{C}^2$ costs satisfying some signature condition on the mixed second derivatives that may include degenerate costs, thus generalizing results previously in the two marginals case and for non-degenerate costs. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic.

Convergence rate of entropy-regularized multi-marginal optimal transport costs

TL;DR

This work analyzes how entropic regularization affects multi-marginal OT costs as the regularization parameter vanishes. It extends prior two-marginal results to the multi-marginal setting, employing a multi-marginal block-approximation and Alexandrov-type arguments to obtain dimension-dependent upper bounds and a signature-based lower bound tied to the mixed second derivatives of the cost. The key contributions include an upper bound for locally Lipschitz costs, a refined bound with a factor for locally semi-concave costs, and a lower bound under a signature condition; in several cases the bounds match, revealing cases where the optimal plan is deterministic. These results clarify how the geometry of marginals and cost curvature govern the convergence rate, with implications for numerical schemes solving entropic MOT problems. Overall, the paper advances understanding of entropy-regularized MOT by linking convergence rates to marginal dimension and Hessian structure rather than the regularized plan itself.

Abstract

We investigate the convergence rate of multi-marginal optimal transport costs that are regularized with the Boltzmann-Shannon entropy, as the noise parameter tends to . We establish lower and upper bounds on the difference with the unregularized cost of the form for some explicit dimensional constants depending on the marginals and on the ground cost, but not on the optimal transport plans themselves. Upper bounds are obtained for Lipschitz costs or locally semi-concave costs for a finer estimate, and lower bounds for costs satisfying some signature condition on the mixed second derivatives that may include degenerate costs, thus generalizing results previously in the two marginals case and for non-degenerate costs. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic.
Paper Structure (7 sections, 4 theorems, 64 equations)

This paper contains 7 sections, 4 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Upper bounds
  4. Upper bound for locally Lipschitz costs
  5. Upper bound for locally semi-concave costs
  6. Lower bound for C2 costs with a signature condition
  7. Examples and matching bound

Key Result

Theorem 1

Let $\gamma$ a solution to pb:mmot and suppose that the signature of some $g\in G_c$ at a point $\bm{x}\in \bm{X}$ is $(d^+,d^-,d^0)$, that is the number of positive, negative and zero eigenvalues. Then, there exists a neighbourhood $N_{\bm{x}}$ of $\bm{x}$ such that $N_{\bm{x}}\bigcap \mathop{\math

Theorems & Definitions (18)

  • Theorem 1: Part of passLocalStructureOptimal2012
  • Remark 2
  • Remark 3: Two marginals case
  • Definition 4: Rényi dimension (following youngDimensionEntropyLyapunov1982)
  • Theorem 5
  • proof
  • Remark 6
  • Definition 7
  • proof
  • Theorem 10
  • ...and 8 more