Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Transport for $ε$-Contaminated Credal Sets: To the Memory of Sayan Mukherjee

Michele Caprio

TL;DR

This work generalizes optimal transport to the setting of lower probabilities, focusing on credal sets characterized by $\epsilon$-contaminations. It introduces concrete formulations for Lower Probability Monge OT (LPM), Lower Probability Kantorovich OT (LPK), and Restricted LPK (RLPK) using the Choquet integral and geometric conditioning, and establishes when these LP problems coincide with their classical counterparts. A key result is that, for $\epsilon$-contaminated credal sets, LPM coincides with classical Monge OT and RLPK coincides with classical Kantorovich OT; however, LPM and LPK need not coincide in general. The paper also provides existence conditions for OT plans under tightness and discusses implications for imprecise probabilistic ML and robust decision-making under uncertainty, with future directions including duality and Brenier-type results.

Abstract

We present generalized versions of Monge's and Kantorovich's optimal transport problems with the probabilities being transported replaced by lower probabilities. We show that, when the lower probabilities are the lower envelopes of $ε$-contaminated sets, then our version of Monge's, and a restricted version of our Kantorovich's problems, coincide with their respective classical versions. We also give sufficient conditions for the existence of our version of Kantorovich's optimal plan, and for the two problems to be equivalent. As a byproduct, we show that for $ε$-contaminations the lower probability versions of Monge's and Kantorovich's optimal transport problems need not coincide. The applications of our results to Machine Learning and Artificial Intelligence are also discussed.

Optimal Transport for $ε$-Contaminated Credal Sets: To the Memory of Sayan Mukherjee

TL;DR

This work generalizes optimal transport to the setting of lower probabilities, focusing on credal sets characterized by -contaminations. It introduces concrete formulations for Lower Probability Monge OT (LPM), Lower Probability Kantorovich OT (LPK), and Restricted LPK (RLPK) using the Choquet integral and geometric conditioning, and establishes when these LP problems coincide with their classical counterparts. A key result is that, for -contaminated credal sets, LPM coincides with classical Monge OT and RLPK coincides with classical Kantorovich OT; however, LPM and LPK need not coincide in general. The paper also provides existence conditions for OT plans under tightness and discusses implications for imprecise probabilistic ML and robust decision-making under uncertainty, with future directions including duality and Brenier-type results.

Abstract

We present generalized versions of Monge's and Kantorovich's optimal transport problems with the probabilities being transported replaced by lower probabilities. We show that, when the lower probabilities are the lower envelopes of -contaminated sets, then our version of Monge's, and a restricted version of our Kantorovich's problems, coincide with their respective classical versions. We also give sufficient conditions for the existence of our version of Kantorovich's optimal plan, and for the two problems to be equivalent. As a byproduct, we show that for -contaminations the lower probability versions of Monge's and Kantorovich's optimal transport problems need not coincide. The applications of our results to Machine Learning and Artificial Intelligence are also discussed.
Paper Structure (11 sections, 12 theorems, 34 equations, 1 figure)

This paper contains 11 sections, 12 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Main Results
  4. Lower Probability Monge's (LPM) Problem
  5. Lower Probability Kantorovich's (LPK) Problem
  6. Restricted Lower Probability Kantorovich's (RLPK) Problem
  7. Equivalence Between LPM And RLPK Problems
  8. Conclusion
  9. On The Choice Of $\underline{P}$
  10. On The Difference Between Conditioning Methods
  11. Proof of Lemma \ref{['well-def']}

Key Result

Lemma 2

The pushforward lower probability $T_\#\underline{P}$ in Definition pf is a well-defined lower probability.

Figures (1)

  • Figure 1: Top: the optimal transport map $T$ between two bell-shaped distributions $P$ and $Q$ on $\mathbb{R}$. Bottom: the optimal transport map $T$ between lower probabilities $\underline{P}$ and $\underline{Q}$ (both depicted as black brushstrokes) that completely characterize credal sets $\mathcal{P}$ and $\mathcal{Q}$ of probabilities on $\mathbb{R}$. The colored distributions are elements of the respective credal sets.

Theorems & Definitions (29)

  • Definition 1: Pushforward Lower Probability, PLP
  • Lemma 2: PLPs are well-defined
  • Definition 3: Choquet Integral
  • Proposition 4: Characterizing the Choquet Integral
  • Corollary 4.1: A Simplification of the Choquet Integral
  • proof
  • Lemma 5: Properties of $\epsilon$-Contaminated Credal Sets
  • proof
  • Lemma 6: A More Convenient Core
  • proof
  • ...and 19 more