pith. machine review for the scientific record. sign in

arxiv: 2604.21745 · v1 · submitted 2026-04-23 · 💻 cs.GL · math.PR

Recognition: unknown

A Brief History of Fr\'echet Distances: From Curves and Probability Laws to FID

Yuli Wu

Authors on Pith no claims yet

Pith reviewed 2026-05-08 12:51 UTC · model grok-4.3

classification 💻 cs.GL math.PR
keywords Fréchet distanceWasserstein distanceFréchet Inception Distanceoptimal transportgenerative model evaluationcurve similarityprobability metricshistory of metric spaces
0
0 comments X

The pith

Fréchet distances link curve geometry to probability couplings and explain the FID score for generative models as a Wasserstein distance between Gaussians.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The note traces Fréchet distances chronologically from the 1906 thesis defining distances on abstract sets, through 1990s algorithms for polygonal curves, to the 1957 coupling-based distance between probability laws. It draws explicit connections between the geometric facet on curves and the distributional facet on laws, relating the latter to Wasserstein distances and optimal transport. The history concludes with the Fréchet Inception Distance used to evaluate deep generative models, where the distance between feature distributions reduces to the closed-form Wasserstein-2 distance between multivariate Gaussians. A reader cares because the account shows how a single classical idea evolved into a practical tool for assessing AI-generated images and other synthetic data.

Core claim

Fréchet's distance, first introduced in 1906 for abstract metric spaces, was later specialized to polygonal curves via the infimum over reparameterizations of the maximum pointwise distance, then extended in 1957 to probability laws as the infimum over couplings of an expected distance that aligns with Wasserstein metrics, and is now applied in FID by modeling feature distributions from generative models as Gaussians whose Wasserstein-2 distance admits an explicit formula.

What carries the argument

The Fréchet distance in its dual geometric form (min over alignments of max deviation between curves) and coupling form (inf over joints of expected distance), which the paper connects to enable the Gaussian-Wasserstein interpretation of FID.

If this is right

  • The 1957 coupling formulation unifies curve similarity and distribution comparison under one infimum operation.
  • FID scores inherit a direct optimal-transport interpretation once features are treated as Gaussians.
  • The provided translations of the 1906 thesis, 1957 paper, and 1950 note make the primary sources available for further study.
  • Computational techniques developed for one facet of the distance may transfer to the other.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Treating sequential data as either curves or empirical measures could yield hybrid metrics that combine geometric and distributional strengths.
  • The lineage suggests FID's effectiveness arises from its grounding in classical metric theory rather than purely empirical construction.
  • Similar historical bridges might exist between other pre-1950 distances and current benchmarks in machine learning.

Load-bearing premise

The historical interpretations and claimed connections between the geometric facet on curves and the distributional facet on probability laws are accurate and add insight beyond existing optimal transport literature.

What would settle it

A side-by-side computation of the geometric Fréchet distance on curve representations versus its coupling-based version on the corresponding empirical measures that yields systematically different values would undermine the claimed equivalence or connection.

Figures

Figures reproduced from arXiv: 2604.21745 by Yuli Wu.

Figure 1
Figure 1. Figure 1: Fréchet’s curve-arcs are ordered and the parameterized trajectories are normalized with view at source ↗
Figure 2
Figure 2. Figure 2: Two curves together with their free-space diagrams for increasing thresholds view at source ↗
Figure 3
Figure 3. Figure 3: Geometric illustration of the CDFs of three one-dimensional Gaussian distributions, view at source ↗
read the original abstract

This note provides a chronological account of Fr\'echet distances, starting with Maurice Fr\'echet's 1906 doctoral thesis on distances in abstract sets and tracing the Fr\'echet distance between polygonal curves and its algorithmic computation in the 1990s. It then continues with his 1957 paper on a coupling-based distance between probability laws with a brief glimpse of Wasserstein distance and optimal transport. We further attempt to draw connections between the distributional, coupling-based facet of Fr\'echet distances on probability laws and the geometric facet on curves. The note ends with a modern use case, the Fr\'echet Inception Distance (FID) in the era of deep generative model evaluation, interpretable as the Wasserstein-2 distance between multivariate Gaussians in a learned feature space. An appendix includes \TeX{}ified faithful English translations of Fr\'echet's 1906 thesis and 1957 paper, and L\'evy's 1950 note for reader convenience.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript provides a chronological historical review of Fréchet distances, beginning with Fréchet's 1906 thesis on distances between curves in abstract metric spaces, moving to the 1957 coupling-based distance between probability laws, sketching links to Wasserstein distance and optimal transport, and concluding with the modern Fréchet Inception Distance (FID) in deep generative model evaluation, which is interpreted as the Wasserstein-2 distance between Gaussians in feature space. An appendix supplies TeX-ified English translations of the 1906 thesis, 1957 paper, and Lévy's 1950 note.

Significance. If the translations prove accurate and the claimed connections between the geometric (curve) and distributional (coupling) facets are substantiated beyond chronology, the note offers a useful synthesis that makes primary sources more accessible and highlights a conceptual bridge relevant to optimal transport and machine-learning evaluation metrics. The inclusion of faithful translations is a concrete strength for readers without direct access to the French originals.

major comments (1)
  1. [§4] §4 (Connections between the geometric and distributional facets): The central interpretive claim—that the 1906 inf-over-reparametrizations sup-norm distance on curves and the 1957 inf-over-couplings expected-distance on laws constitute connected 'facets' of a single concept—rests on descriptive parallels and narrative analogy rather than a shared formal construction (e.g., both as instances of an inf-sup or inf-E schema, or via a cited unifying theorem). This is load-bearing for the paper's stated goal of drawing connections that add insight beyond the name coincidence already noted in the optimal-transport literature.
minor comments (2)
  1. [Abstract and §1] The abstract and §1 refer to 'Fréchet distances' in the plural without an initial clarifying sentence distinguishing the curve metric from the distributional metric; a short definitional paragraph at the outset would improve readability for non-specialists.
  2. [Appendix] Appendix translations: While the provision of primary sources is welcome, the note does not indicate whether the translations were cross-checked against the original French by a second reader or include any footnotes on ambiguous passages; adding a brief translator's note would strengthen scholarly value.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review, positive recommendation for minor revision, and constructive feedback on the manuscript. We address the major comment below and will incorporate revisions accordingly.

read point-by-point responses

  1. Referee: [§4] §4 (Connections between the geometric and distributional facets): The central interpretive claim—that the 1906 inf-over-reparametrizations sup-norm distance on curves and the 1957 inf-over-couplings expected-distance on laws constitute connected 'facets' of a single concept—rests on descriptive parallels and narrative analogy rather than a shared formal construction (e.g., both as instances of an inf-sup or inf-E schema, or via a cited unifying theorem). This is load-bearing for the paper's stated goal of drawing connections that add insight beyond the name coincidence already noted in the optimal-transport literature.

    Authors: We appreciate the referee pointing out the interpretive nature of §4. The manuscript is explicitly framed as a chronological historical review (see abstract: 'attempt to draw connections'), not as a work establishing new formal equivalences or theorems. The parallels noted are structural (both distances arise as infima over classes of transformations: reparametrizations of curves and couplings of measures) together with the shared nomenclature and historical lineage from Fréchet's work. We do not assert a single unifying schema such as a common inf-sup construction or a cited theorem, as none is claimed or needed for the paper's scope. To prevent any possible over-reading, we will revise §4 to (i) explicitly characterize the links as conceptual and historical rather than formal, (ii) describe the shared infimum structure more precisely, and (iii) reference the existing observations of name coincidence in the optimal-transport literature while underscoring the added value of the primary-source translations. These changes will be made in the next version of the manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: purely expository historical account with no derivations or self-referential reductions.

full rationale

The paper is a chronological narrative tracing Fréchet's 1906 and 1957 works, providing translations of primary sources, and offering descriptive parallels between curve and distributional distances. It contains no equations, no fitted parameters, no predictions, and no load-bearing self-citations that reduce claims to inputs. The connections drawn are interpretive and narrative rather than formal derivations that could be circular by construction. This is self-contained against external benchmarks as a history note.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is a historical survey containing no new mathematical derivations, free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5467 in / 1017 out tokens · 23940 ms · 2026-05-08T12:51:31.141348+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

65 extracted references · 36 canonical work pages

  1. [1]

    Alt and M

    H. Alt and M. Godau. Measuring the resemblance of polygonal curves. InProceedings of the eighth annual symposium on Computational geometry, pages 102–109, 1992.DOI:10.1145/142675.142699. 2, 6, 26

    work page doi:10.1145/142675.142699 1992

  2. [2]

    Alt and M

    H. Alt and M. Godau. Computing the Fréchet distance between two polygonal curves.International Journal of Computational Geometry & Applications, 5(01n02):75–91, 1995.DOI: 10.1142/S0218195995000064. 2, 6, 26

    work page doi:10.1142/s0218195995000064 1995

  3. [3]

    Barbut, B

    M. Barbut, B. Locker, and L. Mazliak.Paul Lévy and Maurice Fréchet: 50 Years of Correspondence in 107 Letters. Springer, 2013.DOI:10.1007/978-1-4471-5619-2. 2, 24, 108

    work page doi:10.1007/978-1-4471-5619-2 2013

  4. [4]

    Bhattacharyya

    A. Bhattacharyya. On a measure of divergence between two statistical populations defined by their probability distribution.Bulletin of the Calcutta Mathematical Society, 35:99–110, 1943. 23

    1943

  5. [5]

    Billingsley.Convergence of Probability Measures

    P. Billingsley.Convergence of Probability Measures. John Wiley & Sons, 2 edition, 1999.DOI: 10.1002/ 9780470316962. 23, 24, 25

    1999

  6. [6]

    Bińkowski, D

    M. Bińkowski, D. J. Sutherland, M. Arbel, and A. Gretton. Demystifying MMD GANs. InInternational Conference on Learning Representations, 2018. 20, 21, 25

    2018

  7. [7]

    [Bri19] Karl Bringmann

    K. Bringmann. Why walking the dog takes time: Fréchet distance has no strongly subquadratic algorithms unless SETH fails. In2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 661–670. IEEE, 2014.DOI:10.1109/FOCS.2014.76. 2

    work page doi:10.1109/focs.2014.76 2014

  8. [8]

    Buchin, M

    K. Buchin, M. Buchin, W. Meulemans, and W. Mulzer. Four soviets walk the dog: Improved bounds for computing the Fréchet distance.Discrete & Computational Geometry, 58(1):180–216, 2017.DOI: 10.1007/s00454-017-9878-7. 2, 6

    work page doi:10.1007/s00454-017-9878-7 2017

  9. [9]

    Buchin, T

    K. Buchin, T. Ophelders, and B. Speckmann. SETH says: Weak Fréchet distance is faster, but only if it is continuous and in one dimension. InProceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2887–2901. SIAM, 2019.DOI:10.5555/3458064.3458135. 2

    work page doi:10.5555/3458064.3458135 2019

  10. [10]

    Champion, L

    T. Champion, L. De Pascale, and P. Juutinen. The∞-Wasserstein distance: Local solutions and existence of optimal transport maps.SIAM Journal on Mathematical Analysis, 40(1):1–20, 2008.DOI: 10.1137/ 07069938X. 15, 17

    2008

  11. [11]

    Cheng, H

    S.-W. Cheng, H. Huang, and S. Zhang. Constant approximation of Fréchet distance in strongly subquadratic time. InProceedings of the 57th Annual ACM Symposium on Theory of Computing, pages 2329–2340, 2025. DOI:10.1145/3717823.3718157. 2

    work page doi:10.1145/3717823.3718157 2025

  12. [12]

    M. J. Chong and D. Forsyth. Effectively unbiased FID and inception score and where to find them. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6070–6079, 2020.DOI:10.1109/CVPR42600.2020.00611. 21 27

    work page doi:10.1109/cvpr42600.2020.00611 2020

  13. [13]

    Conradi, A

    J. Conradi, A. Driemel, and B. Kolbe. Revisiting the Fréchet distance between piecewise smooth curves. Computational Geometry, 129:102194, 2025.DOI:10.1016/j.comgeo.2025.102194. 2

    work page doi:10.1016/j.comgeo.2025.102194 2025

  14. [14]

    D. C. Dowson and B. V. Landau. The Fréchet distance between multivariate normal distributions.Journal of Multivariate Analysis, 12(3):450–455, 1982.DOI:10.1016/0047-259X(82)90077-X. 15, 16, 26

    work page doi:10.1016/0047-259x(82)90077-x 1982

  15. [15]

    G. M. Ewing.Calculus of Variations with Applications. W. W. Norton & Company, 1969.URL: https: //archive.org/details/calculusofvariat0000ewin. 6

    1969

  16. [16]

    M. Fréchet. Sur quelques points du calcul fonctionnel.Rendiconti del Circolo Matematico di Palermo, 22 (1):1–72, 1906.DOI:10.1007/BF03018603. 2, 3, 4, 5, 6, 8, 10, 11, 26

    work page doi:10.1007/bf03018603 1906

  17. [17]

    Fréchet.Recherches théoriques modernes sur le calcul des probabilités: livre

    M. Fréchet.Recherches théoriques modernes sur le calcul des probabilités: livre. Méthode des fonctions arbitraires. Théorie des événements en chaîne dans le cas d’un nombre fini d’états possibles. Avec supplément nouveau et une note de P. Lévy. Gauthier-Villars, 1950. 2, 10, 17, 26

    1950

  18. [18]

    M. Fréchet. La kanonaj formoj de la 2, 3, 4 - dimensiaj paraanalitikaj funkcioj.Compositio Mathematica, 12:81–96, 1954-1956.URL:https://www.numdam.org/item/CM_1954-1956__12__81_0/. 108

    1954

  19. [19]

    M. Fréchet. Sur la distance de deux lois de probabilité. InAnnales de l’ISUP, volume 6, pages 183–198, 1957.URL:https://hal.science/hal-04093677v1. 2, 10, 11, 12, 14, 15, 26

    1957

  20. [20]

    Computational optimal transport.Found

    P. Gabriel and C. Marco. Computational optimal transport with applications to data sciences.Foundations and Trends®in Machine Learning, 11(5-6):355–607, 2019.DOI: 10.1561/2200000073. 7, 8, 12, 13, 14, 15, 16, 18, 21, 22, 23, 24, 25, 26

    work page doi:10.1561/2200000073 2019

  21. [21]

    Gelbrich

    M. Gelbrich. On a formula for the L2 Wasserstein metric between measures on Euclidean and Hilbert spaces.Mathematische Nachrichten, 147(1):185–203, 1990.DOI:10.1002/mana.19901470121. 16

    work page doi:10.1002/mana.19901470121 1990

  22. [22]

    Genevay, G

    A. Genevay, G. Peyré, and M. Cuturi. Learning generative models with Sinkhorn divergences. In Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, pages 1608–1617, 2018. 25

    2018

  23. [23]

    A. L. Gibbs and F. E. Su. On choosing and bounding probability metrics.International Statistical Review, 70(3):419–435, 2002.DOI:10.1111/j.1751-5823.2002.tb00178.x. 21

    work page doi:10.1111/j.1751-5823.2002.tb00178.x 2002

  24. [24]

    M. Godau. A natural metric for curves—computing the distance for polygonal chains and approximation algorithms. InAnnual Symposium on Theoretical Aspects of Computer Science, pages 127–136. Springer, 1991.DOI:10.1007/BFb0020793. 6

    work page doi:10.1007/bfb0020793 1991

  25. [25]

    Gretton, K

    A. Gretton, K. Borgwardt, M. Rasch, B. Schölkopf, and A. Smola. A kernel method for the two-sample- problem.Advances in Neural Information Processing Systems, 19:513–520, 2006. 24

    2006

  26. [26]

    Gutschlag and S

    T. Gutschlag and S. Storandt. On the generalized fréchet distance and its applications. InProceedings of the 30th International Conference on Advances in Geographic Information Systems, pages 1–10, 2022.DOI: 10.1145/3557915.3560970. 2

    work page doi:10.1145/3557915.3560970 2022

  27. [27]

    Hausdorff.Grundzüge der Mengenlehre

    F. Hausdorff.Grundzüge der Mengenlehre. Veit & Comp., Leipzig, 1914. 23

    1914

  28. [28]

    Hellinger

    E. Hellinger. Neue begründung der theorie quadratischer formen von unendlichvielen veränderlichen. Journal für die reine und angewandte Mathematik, 136:210–271, 1909.DOI: 10.1515/crll.1909.136

    work page doi:10.1515/crll.1909.136 1909

  29. [29]

    Heusel, H

    M. Heusel, H. Ramsauer, T. Unterthiner, B. Nessler, and S. Hochreiter. GANs trained by a two time-scale update rule converge to a local Nash equilibrium.Advances in Neural Information Processing Systems, 30: 6626–6637, 2017. 2, 16, 20, 26

    2017

  30. [30]

    D. Hu, L. Chen, H. Fang, Z. Fang, T. Li, and Y. Gao. Spatio-temporal trajectory similarity measures: A comprehensive survey and quantitative study.IEEE Transactions on Knowledge and Data Engineering, 36 (5):2191–2212, 2023.DOI:10.1109/TKDE.2023.3323535. 21

    work page doi:10.1109/tkde.2023.3323535 2023

  31. [31]

    Jayasumana, S

    S. Jayasumana, S. Ramalingam, A. Veit, D. Glasner, A. Chakrabarti, and S. Kumar. Rethinking FID: Towards a better evaluation metric for image generation. InProceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9307–9315, 2024.DOI:10.1109/CVPR52733.2024.00889. 21

    work page doi:10.1109/cvpr52733.2024.00889 2024

  32. [32]

    A. N. Kolmogorov. Sulla determinazione empirica di una legge di distribuzione.Giornale dell’Istituto Italiano degli Attuari, 4:83–91, 1933. 24

    1933

  33. [33]

    The Annals of Mathematical Statistics , author =

    S. Kullback and R. A. Leibler. On information and sufficiency.The Annals of Mathematical Statistics, 22(1): 79–86, 1951.DOI:10.1214/aoms/1177729694. 22 28

    work page doi:10.1214/aoms/1177729694 1951

  34. [34]

    Kynkäänniemi, T

    T. Kynkäänniemi, T. Karras, S. Laine, J. Lehtinen, and T. Aila. Improved precision and recall metric for assessing generative models. InAdvances in Neural Information Processing Systems, volume 32, pages 3927–3936, 2019. 21

    2019

  35. [35]

    J. Lin. Divergence measures based on the Shannon entropy.IEEE Transactions on Information Theory, 37 (1):145–151, 1991.DOI:10.1109/18.61115. 22

    work page doi:10.1109/18.61115 1991

  36. [36]

    Majumdar and V

    R. Majumdar and V. S. Prabhu. Computing the Skorokhod distance between polygonal traces. In Proceedings of the 18th International Conference on Hybrid Systems: Computation and Control, pages 199–208, 2015.DOI:10.1145/2728606.2728618. 25

    work page doi:10.1145/2728606.2728618 2015

  37. [37]

    M. Morse. Functional topology and abstract variational theory.Mémorial des Sciences Mathématiques, 92: 1–50, 1938.URL:https://www.jstor.org/stable/87124. 5

    1938

  38. [38]

    M. F. Naeem, S. J. Oh, Y. Uh, Y. Choi, and J. Yoo. Reliable fidelity and diversity metrics for generative models. InProceedings of the 37th International Conference on Machine Learning, pages 7176–7185, 2020. 21

    2020

  39. [39]

    Parmar, R

    G. Parmar, R. Zhang, and J.-Y. Zhu. On aliased resizing and surprising subtleties in GAN evaluation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 11410–11420, 2022. DOI:10.1109/CVPR52688.2022.01112. 20, 21

    work page doi:10.1109/cvpr52688.2022.01112 2022

  40. [40]

    Y. V. Prokhorov. Convergence of random processes and limit theorems in probability theory.Theory of Probability and Its Applications, 1(2):157–214, 1956.DOI:10.1137/1101016. 24

    work page doi:10.1137/1101016 1956

  41. [41]

    Rüschendorf

    L. Rüschendorf. Wasserstein metric. In M. Hazewinkel, editor,Encyclopaedia of Mathematics: Supplement Volume II, pages 487–488. Springer Netherlands, 2000.DOI:10.1007/978-94-015-1279-4_23. 26

    work page doi:10.1007/978-94-015-1279-4_23 2000

  42. [42]

    M. S. M. Sajjadi, O. Bachem, M. Lucic, O. Bousquet, and S. Gelly. Assessing generative models via precision and recall. InAdvances in Neural Information Processing Systems, volume 31, pages 5228–5237, 2018. 21

    2018

  43. [43]

    Sakoe and S

    H. Sakoe and S. Chiba. Dynamic programming algorithm optimization for spoken word recognition. IEEE Transactions on Acoustics, Speech, and Signal Processing, 26(1):43–49, 1978.DOI: 10.1109/TASSP. 1978.1163055. 21

    work page doi:10.1109/tassp 1978

  44. [44]

    Santambrogio.Optimal Transport for Applied Mathematicians

    F. Santambrogio.Optimal Transport for Applied Mathematicians. Birkhäuser, 2015.DOI: 10.1007/ 978-3-319-20828-2. 7, 12, 15, 17, 18

    2015

  45. [45]

    A. V. Skorokhod. Limit theorems for stochastic processes.Theory of Probability and Its Applications, 1(3): 261–290, 1956.DOI:10.1137/1101022. 25

    work page doi:10.1137/1101022 1956

  46. [46]

    M. A. Stephens.Introduction to Kolmogorov (1933) On the Empirical Determination of a Distribution, pages 93–105. Springer New York, 1992.DOI:10.1007/978-1-4612-4380-9_9. 24

    work page doi:10.1007/978-1-4612-4380-9_9 1933

  47. [47]

    G. J. Székely and M. L. Rizzo. Energy statistics: A class of statistics based on distances.Journal of Statistical Planning and Inference, 143(8):1249–1272, 2013.DOI:10.1016/j.jspi.2013.03.018. 25

    work page doi:10.1016/j.jspi.2013.03.018 2013

  48. [48]

    A. E. Taylor. A study of Maurice Fréchet: I. His early work on point set theory and the theory of functionals.Archive for History of Exact Sciences, 27(3):233–295, 1982.DOI: 10.1007/BF00327860. 2, 3, 108

    work page doi:10.1007/bf00327860 1982

  49. [49]

    A. E. Taylor. A study of Maurice Fréchet: II. Mainly about his work on general topology, 1909–1928. Archive for History of Exact Sciences, 34(4):279–380, 1985.DOI:10.1007/BF00411640. 2, 108

    work page doi:10.1007/bf00411640 1909

  50. [50]

    A. E. Taylor. A study of Maurice Fréchet: III. Fréchet as analyst, 1909–1930.Archive for History of Exact Sciences, 37(1):25–76, 1987.DOI:10.1007/BF00412329. 2, 108

    work page doi:10.1007/bf00412329 1909

  51. [51]

    A. E. Taylor, P. Dugac, and H. Lebesgue. Quatre lettres de Lebesgue à Fréchet.Revue d’Histoire des Sciences, pages 149–169, 1981.DOI:10.3406/rhs.1981.1861. 14, 108

    work page doi:10.3406/rhs.1981.1861 1981

  52. [52]

    Vallender

    S. Vallender. Calculation of the Wasserstein distance between probability distributions on the line.Theory of Probability & Its Applications, 18(4):784–786, 1974.DOI:10.1137/1118101. 33

    work page doi:10.1137/1118101 1974

  53. [53]

    van Dijk

    T. van Dijk. Fréchet distance: Free-space diagram, 2026.URL: https://www1.pub.informatik. uni-wuerzburg.de/pub/dijk/frechet/. Accessed: 2026-04-08. 7

    2026

  54. [54]

    L. N. Vaserstein. Markov processes over denumerable products of spaces, describing large systems of automata.Problemy Peredachi Informatsii, 5(3):64–72, 1969. 12 29

    1969

  55. [55]

    Villani et al.Optimal transport: Old and new, volume 338

    C. Villani et al.Optimal transport: Old and new, volume 338. Springer, 2009.DOI: 10.1007/ 978-3-540-71050-9. 7, 12

    2009

  56. [56]

    A. J. Ward. A generalization of the Fréchet distance of two curves.Proceedings of the National Academy of Sciences, 40(6):519–521, 1954.DOI:10.1073/pnas.40.7.598. 3, 5

    work page doi:10.1073/pnas.40.7.598 1954

  57. [57]

    A. J. Ward. A second generalization of the Fréchet distance of two curves.Proceedings of the National Academy of Sciences, 40(7):609–612, 1954.DOI:10.1073/pnas.40.10.1011. 5

    work page doi:10.1073/pnas.40.10.1011 1954

  58. [58]

    Fréchet distance — Wikipedia, the free encyclopedia, 2026.URL: https:// en.wikipedia.org/w/index.php?title=Fr%C3%A9chet_distance&oldid=1331641232

    Wikipedia contributors. Fréchet distance — Wikipedia, the free encyclopedia, 2026.URL: https:// en.wikipedia.org/w/index.php?title=Fr%C3%A9chet_distance&oldid=1331641232. [Online; accessed 23-March-2026]. 2

    2026

  59. [59]

    distance

    Y. Wu, F. Liu, R. Yilmaz, H. Konermann, P. Walter, and J. Stegmaier. A pragmatic note on evaluating generative models with Fréchet inception distance for retinal image synthesis. InMedical Imaging with Deep Learning, 2026.URL:https://openreview.net/forum?id=NabZo8HWuh. 21 30 A Proofs A.1 Extremal bounds for admissible bivariate laws Let𝐹and𝐺be one-dimensi...

    2026

  60. [60]

    Moreover, if𝐸is extremal, convergence is necessarily uniform on𝐸

    1st Lemma.— The limit 𝑈of a convergent sequence𝑈1,𝑈2,…,𝑈𝑛,…of operations equally continuous on a set𝐸is a continuous operation on𝐸. Moreover, if𝐸is extremal, convergence is necessarily uniform on𝐸. Indeed, if one considers an arbitrary element 𝐴of 𝐸, limit of a sequence of elements of 𝐸: 𝐴1,𝐴2,…,𝐴𝑛,…, one has ||𝑈𝑞(𝐴)−𝑈𝑞(𝐴𝑛)||<𝜀,for𝑛>𝑝, whatever𝑞may be. If...

  61. [61]

    If these operations are equally continuous on set𝐹formed by elements of𝐷and of its derived set𝐷′, the sequence considered is also convergent on𝐹

    2nd Lemma.— Consider a sequence of operations 𝑈1,𝑈2,…convergent on a set𝐷. If these operations are equally continuous on set𝐹formed by elements of𝐷and of its derived set𝐷′, the sequence considered is also convergent on𝐹. It suffices to prove that the sequence is convergent at every element𝐴that is limit of a sequence of elements of𝐷:𝐴1,𝐴2,…,𝐴𝑛,…Now, given...

  62. [62]

    enumerates

    At every point common to these two sets one has |||𝑓−𝑓(𝑟) 𝑞 |||<𝜀, and set of these common points has measure at least equal to 1−( 𝜎 2+ 𝜎 2)=1−𝜎. A fortiori, the same is true of the set of points such that|||𝑓−𝑓(𝑟) 𝑞 |||<𝜀. This being so, let 𝜀and𝜎take two successive sequences of values 𝜀1,𝜀2,…;𝜎1,𝜎2,…Whatever𝑟, one can determine integers𝑛𝑟,𝑝𝑟such that s...

    1905

  63. [63]

    One can therefore find a number𝑞such that their sum is less than 𝜀 2 for𝑛>𝑞

    With 𝑝thusfixed, it is clear that the sum of the 𝑝preceding terms in inequality (2) tends to zero as𝑛increases indefinitely. One can therefore find a number𝑞such that their sum is less than 𝜀 2 for𝑛>𝑞. In summary, we obtain a number𝑞such that: (𝑥(𝑛),𝑥)<𝜀,for𝑛>𝑞; which proves that the assumed condition was indeed sufficient. Let us prove it is necessary. T...

  64. [64]

    This being so, let 𝜆be any number between𝑢0 and𝑢′ 0; for𝑛large enough,𝜑𝑝𝑛(𝑡𝑛) and𝜑𝑝𝑛(𝑡′ 𝑛), which tend to𝑢0 and𝑢′ 0, will contain𝜆between them

    Then inequalities (7) show that 𝑡′ 𝑛also tends to 𝜏0 and that |𝑢0 −𝑢′ 0|≥𝜀. This being so, let 𝜆be any number between𝑢0 and𝑢′ 0; for𝑛large enough,𝜑𝑝𝑛(𝑡𝑛) and𝜑𝑝𝑛(𝑡′ 𝑛), which tend to𝑢0 and𝑢′ 0, will contain𝜆between them. The continuous function𝜑𝑝𝑛(𝑡)therefore takes the value𝜆at least once for a value𝜏𝑛of𝑡between𝑡𝑛and𝑡′ 𝑛. One therefore has, by (6), |𝑓(𝜏𝑛)−...

  65. [65]

    between two probability laws,

    We thus reach the announced contradiction, since 𝐹(𝑢),𝐺(𝑢),𝐻(𝑢)are not simultaneously constant on any interval. One should make an exception for the case where𝚪reduces to a point, but then the converse is obvious. 80.To show that condition b) of n◦49 is satisfied, consider three curves 𝛾, 𝚪, and 𝐶, having representations (4), (5), and 𝑥=𝑎(𝑣), 𝑦=𝑏(𝑣), 𝑧=𝑐(...

    1906