### Abstract

We present an algebraic account of the Wasserstein distances W_{p} on complete metric spaces, for p ≥ 1. This is part of a program of a quantitative algebraic theory of effects in programming languages. In particular, we give axioms, parametric in p, for algebras over metric spaces equipped with probabilistic choice operations. The axioms say that the operations form a barycentric algebra and that the metric satisfies a property typical of the Wasserstein distance W_{p}. We show that the free complete such algebra over a complete metric space is that of the Radon probability measures with finite moments of order p, equipped with the Wasserstein distance as metric and with the usual binary convex sums as operations.

Language | English |
---|---|

Article number | 19 |

Number of pages | 16 |

Journal | Logical Methods in Computer Science |

Volume | 14 |

Issue number | 3 |

DOIs | |

Publication status | Published - 14 Sep 2018 |

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### Keywords

- Wasserstein distances
- quantitative algebraic theory
- programming languages

### Cite this

*Logical Methods in Computer Science*,

*14*(3), [19]. https://doi.org/10.23638/LMCS-14(3:19)2018

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*Logical Methods in Computer Science*, vol. 14, no. 3, 19. https://doi.org/10.23638/LMCS-14(3:19)2018

**Free complete Wasserstein algebras.** / Mardare, Radu; Panangaden, Prakash; Plotkin, Gordon D.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Free complete Wasserstein algebras

AU - Mardare, Radu

AU - Panangaden, Prakash

AU - Plotkin, Gordon D.

PY - 2018/9/14

Y1 - 2018/9/14

N2 - We present an algebraic account of the Wasserstein distances Wp on complete metric spaces, for p ≥ 1. This is part of a program of a quantitative algebraic theory of effects in programming languages. In particular, we give axioms, parametric in p, for algebras over metric spaces equipped with probabilistic choice operations. The axioms say that the operations form a barycentric algebra and that the metric satisfies a property typical of the Wasserstein distance Wp. We show that the free complete such algebra over a complete metric space is that of the Radon probability measures with finite moments of order p, equipped with the Wasserstein distance as metric and with the usual binary convex sums as operations.

AB - We present an algebraic account of the Wasserstein distances Wp on complete metric spaces, for p ≥ 1. This is part of a program of a quantitative algebraic theory of effects in programming languages. In particular, we give axioms, parametric in p, for algebras over metric spaces equipped with probabilistic choice operations. The axioms say that the operations form a barycentric algebra and that the metric satisfies a property typical of the Wasserstein distance Wp. We show that the free complete such algebra over a complete metric space is that of the Radon probability measures with finite moments of order p, equipped with the Wasserstein distance as metric and with the usual binary convex sums as operations.

KW - Wasserstein distances

KW - quantitative algebraic theory

KW - programming languages

UR - http://www.scopus.com/inward/record.url?scp=85055831612&partnerID=8YFLogxK

U2 - 10.23638/LMCS-14(3:19)2018

DO - 10.23638/LMCS-14(3:19)2018

M3 - Article

VL - 14

JO - Logical Methods in Computer Science

T2 - Logical Methods in Computer Science

JF - Logical Methods in Computer Science

SN - 1860-5974

IS - 3

M1 - 19

ER -