Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm

TL;DR

This work addresses entropic optimal transport in unbounded settings by extending Hilbert's projective metric to cones of functions with bounded growth, enabling contractivity analyses for kernel operators without requiring bounded costs. The authors introduce tail-controlled test-function cones $\mathcal{F}^{\mu}_{\alpha,\tilde{\alpha}}$ and their duals $\mathcal{G}^{\mu}_{\alpha,\tilde{\alpha}}$, define the corresponding Hilbert metric $d_{\mathcal{G}^{\mu}_{\alpha,\tilde{\alpha}}}$, and prove kernel operators $L_{K,\mu}$ are contractions under suitable decay conditions on $K$ and the marginals. They then establish exponential convergence of Sinkhorn's algorithm in this unbounded regime under tail-growth assumptions on the cost and marginals, with explicit rates $\kappa<1$ and a bound $A$ such that $d_{\mathcal{G}}(g_1^{(n)}, g_1^*) + d_{\mathcal{G}}(g_2^{(n)}, g_2^*) \le A\kappa^n$ and $\|\pi^{(n)}-\pi^*\|_{TV} \le A\kappa^n$. The results generalize beyond bounded-cost settings and yield exponential convergence of primal Sinkhorn iterations in total variation via product-space embeddings. Overall, the paper provides new, tail-aware contraction tools for Hilbert's metric that unlock robust, quantitative guarantees for entropic OT in practical, unbounded scenarios.

Abstract

Motivated by the entropic optimal transport problem in unbounded settings, we study versions of Hilbert's projective metric for spaces of integrable functions of bounded growth. These versions of Hilbert's metric originate from cones which are relaxations of the cone of all non-negative functions, in the sense that they include all functions having non-negative integral values when multiplied with certain test functions. We show that kernel integral operators are contractions with respect to suitable specifications of such metrics even for kernels which are not bounded away from zero, provided that the decay to zero of the kernel is controlled. As an application to entropic optimal transport, we show exponential convergence of Sinkhorn's algorithm in settings where the marginal distributions have sufficiently light tails compared to the growth of the cost function.

Figures (2)

• Figure 1: The left hand side depicts a simple setting with a light-tailed measure, $\mu \sim |X|$, $X$ standard normal, and we wish to compare two functions of bounded growth, $g$ and $\tilde{g}$. The right hand image shows the start of a sequence of functions $f_1, f_2, f_3, \dots$ which show that formula \ref{['eq:dGdef']} for the baseline version of Hilbert's metric evaluates to $d_{\mathcal{G}}(g, \tilde{g}) = \infty$.
• Figure 2: The left hand side shows how condition \ref{['eq:conditionintuition']} alleviates the undesired behavior in the example of Figure \ref{['fig:example1']}. With condition \ref{['eq:conditionintuition']}, the sequence of functions shown in the right panel in Figure \ref{['fig:example1']} is not a valid sequence of test functions, since already $f_3$ puts a larger fraction of its mass into the tails than allowed by the function $\alpha$. The right hand side shows a function $f$ satisfying condition \ref{['eq:conditionintuition']}. This function illustrates that condition \ref{['eq:conditionintuition']} still allows test function to have positive values in the tails, as long as those are not the only positive values or disproportionately large.

Theorems & Definitions (44)

• Theorem 1.1
• proof
• Theorem 1.2
• proof
• Theorem 2.1: Birkhoff's contraction theorem birkhoff1957extensions
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof