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Transport f divergences

Wuchen Li

TL;DR

This work introduces transport $f$-divergences, a one-dimensional generalization of information divergences built from monotone transport maps that push one density to another. By combining convex $f$ with transport-derived Jacobians, the authors derive multiple equivalent formulations, variational duals, and small-step Taylor expansions in Wasserstein-2 geometry, connecting classical $f$-divergences with optimal transport. They present a suite of concrete examples, including transport KL, transport TV, transport Hessian distances, and $oldsymbol{\alpha}$-divergences, and provide analytical expressions in location-scale and generative settings. The framework offers a geometry-aware alternative to standard divergences with potential applications in sampling, generative modeling, and Bayesian inverse problems, and lays out directions for extending to higher dimensions via Jacobian matrices and associated inequalities and algorithms.

Abstract

We define a class of divergences to measure differences between probability density functions in one-dimensional sample space. The construction is based on the convex function with the Jacobi operator of mapping function that pushforwards one density to the other. We call these information measures transport f-divergences. We present several properties of transport $f$-divergences, including invariances, convexities, variational formulations, and Taylor expansions in terms of mapping functions. Examples of transport f-divergences in generative models are provided.

Transport f divergences

TL;DR

This work introduces transport -divergences, a one-dimensional generalization of information divergences built from monotone transport maps that push one density to another. By combining convex with transport-derived Jacobians, the authors derive multiple equivalent formulations, variational duals, and small-step Taylor expansions in Wasserstein-2 geometry, connecting classical -divergences with optimal transport. They present a suite of concrete examples, including transport KL, transport TV, transport Hessian distances, and -divergences, and provide analytical expressions in location-scale and generative settings. The framework offers a geometry-aware alternative to standard divergences with potential applications in sampling, generative modeling, and Bayesian inverse problems, and lays out directions for extending to higher dimensions via Jacobian matrices and associated inequalities and algorithms.

Abstract

We define a class of divergences to measure differences between probability density functions in one-dimensional sample space. The construction is based on the convex function with the Jacobi operator of mapping function that pushforwards one density to the other. We call these information measures transport f-divergences. We present several properties of transport -divergences, including invariances, convexities, variational formulations, and Taylor expansions in terms of mapping functions. Examples of transport f-divergences in generative models are provided.

Paper Structure

This paper contains 11 sections, 6 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Review of $f$-divergences
  3. Transport $f$-divergences
  4. Review of Wasserstein-$2$ distances
  5. Transport $f$-divergences
  6. Properties
  7. Variational formulations
  8. Local behaviors and Taylor expansions
  9. Examples
  10. Examples
  11. Discussion

Key Result

Proposition 1

The following equivalent formulations hold:

Theorems & Definitions (21)

  • Definition 1: Transport $f$-divergence
  • Proposition 1: Equivalent formulations
  • proof
  • Proposition 2: Properties
  • proof
  • Theorem 1: Transport $f$-dualities
  • proof
  • Theorem 2: Transport $f$-Kantorvich dualities
  • proof
  • Theorem 3: Local behaviors
  • ...and 11 more