Lifting couplings in Wasserstein spaces

TL;DR

This paper establishes a mathematical framework connecting conditional probabilities, transport plans, and geometric concepts like lenses and submetries within Wasserstein spaces, revealing new insights into their structure and relationships.

Contribution

It introduces a category-theoretic and geometric framework for understanding couplings and conditional probabilities as liftings in Wasserstein spaces, unifying concepts across probability, geometry, and category theory.

Findings

Weighted categories of probability measures relate to submetry in geometry.

Wasserstein spaces can be modeled as pseudo-quasimetric spaces derived from weighted categories.

Conditional probabilities can be viewed as liftings that preserve transport plan costs.

Abstract

This paper makes mathematically precise the idea that conditional probabilities are analogous to path liftings in geometry. The idea of lifting is modelled in terms of the category-theoretic concept of a lens, which can be interpreted as a consistent choice of arrow liftings. The category we study is the one of probability measures over a given standard Borel space, with morphisms given by the couplings, or transport plans. The geometrical picture is even more apparent once we equip the arrows of the category with weights, which one can interpret as "lengths" or "costs", forming a so-called weighted category, which unifies several concepts of category theory and metric geometry. Indeed, we show that the weighted version of a lens is tightly connected to the notion of submetry in geometry. Every weighted category gives rise to a pseudo-quasimetric space via optimization over the arrows. In particular, Wasserstein spaces can be obtained from the weighted categories of probability measures and their couplings, with the weight of a coupling given by its cost. In this case, conditionals allow one to form weighted lenses, which one can interpret as "lifting transport plans, while preserving their cost".