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Lifting couplings in Wasserstein spaces

Paolo Perrone

TL;DR

The paper develops a unified framework linking conditional probabilities to liftings in geometry using lenses within weighted categories of probability measures. By treating couplings as morphisms and costs as weights, Wasserstein spaces arise as optimization over arrows, and conditioning yields liftings that preserve both composition and cost. The main contribution is a functorial construction: measurable lenses between base spaces induce weighted lenses between their coupling categories, providing a principled method to lift transport plans along point liftings, with explicit formulas and preservation of Wasserstein costs. This formalism bridges category theory, metric geometry, and optimal transport, enabling systematic lifting of transport plans and offering a robust, compositional view of conditional distributions in geometric spaces with 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".

Lifting couplings in Wasserstein spaces

TL;DR

The paper develops a unified framework linking conditional probabilities to liftings in geometry using lenses within weighted categories of probability measures. By treating couplings as morphisms and costs as weights, Wasserstein spaces arise as optimization over arrows, and conditioning yields liftings that preserve both composition and cost. The main contribution is a functorial construction: measurable lenses between base spaces induce weighted lenses between their coupling categories, providing a principled method to lift transport plans along point liftings, with explicit formulas and preservation of Wasserstein costs. This formalism bridges category theory, metric geometry, and optimal transport, enabling systematic lifting of transport plans and offering a robust, compositional view of conditional distributions in geometric spaces with 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".

Paper Structure

This paper contains 18 sections, 12 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Projections and liftings.
  3. Optimal transport and couplings.
  4. Cost functions, distances, and weights.
  5. Outline.
  6. Acknowledgements.
  7. Weighted categories
  8. Main definitions
  9. Isomorphisms
  10. Embeddings
  11. Lenses and submetries
  12. Lifting points
  13. Lifting paths
  14. The category of couplings
  15. Composition
  16. ...and 3 more sections

Key Result

Proposition 2.16

Let $F:{\mathpzc{C}}\to{\mathpzc{D}}$ be an embedding of weighted categories, and let $X$ and $Y$ be objects of ${\mathpzc{C}}$. If $FX$ and $FY$ are isomorphic (resp. quasi-isomorphic) in ${\mathpzc{D}}$, then $X$ and $Y$ are already isomorphic (resp. quasi-isomorphic) in ${\mathpzc{C}}$.

Theorems & Definitions (49)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 39 more