IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v37y2024i4d10.1007_s10959-024-01349-x.html
   My bibliography  Save this article

The Transport Map Computed by Iterated Function System

Author

Listed:
  • Judy Yangjun Lin

    (Soochow University)

  • Huoxia Liu

    (Zhejiang A &F University)

Abstract

The transport map in the optimal transport model plays an important role in the machine learning and statistics fields, and the approximation of the transport map is significant for application. Since the transport map has no explicit expression in the general case, representations of such a map or realizing its action are often intractable as the dimension increases. In this paper, we adopt a new perspective to approximate the transport map by using an iterated function system: the transport map is constructed through a composition of some iterated maps. The source measure in the optimal transport model is transferred through the iterated maps, and the corresponding sequence of iterated measures has been produced. We show that there is an $$\epsilon $$ ϵ -optimal transport map between the iterated measure and target measure in each iteration of the iterated function system. Theoretical convergence analysis of the sequence of iterated measures is performed, the iterated measures converge to the target measure in the optimal transport model, and we get a geometric convergence rate if the iterated maps are Lipschitz continuous and independent from each other. Finally, we give the statistical estimation error of the transport map approximated by the iterated function system.

Suggested Citation

  • Judy Yangjun Lin & Huoxia Liu, 2024. "The Transport Map Computed by Iterated Function System," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3725-3755, November.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:4:d:10.1007_s10959-024-01349-x
    DOI: 10.1007/s10959-024-01349-x
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-024-01349-x
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-024-01349-x?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Karren Dai Yang & Karthik Damodaran & Saradha Venkatachalapathy & Ali C Soylemezoglu & G V Shivashankar & Caroline Uhler, 2020. "Predicting cell lineages using autoencoders and optimal transport," PLOS Computational Biology, Public Library of Science, vol. 16(4), pages 1-20, April.
    2. Roberts, Gareth O. & Rosenthal, Jeffrey S., 2002. "One-shot coupling for certain stochastic recursive sequences," Stochastic Processes and their Applications, Elsevier, vol. 99(2), pages 195-208, June.
    3. Jeremy Heng & Arnaud Doucet & Yvo Pokern, 2021. "Gibbs flow for approximate transport with applications to Bayesian computation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(1), pages 156-187, February.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jovanovski, Oliver, 2014. "Convergence bound in total variation for an image restoration model," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 11-16.
    2. A. Beskos & G. O. Roberts, 2005. "One-Shot CFTP; Application to a Class of Truncated Gaussian Densities," Methodology and Computing in Applied Probability, Springer, vol. 7(4), pages 407-437, December.
    3. Xinyi Zhang & Xiao Wang & G. V. Shivashankar & Caroline Uhler, 2022. "Graph-based autoencoder integrates spatial transcriptomics with chromatin images and identifies joint biomarkers for Alzheimer’s disease," Nature Communications, Nature, vol. 13(1), pages 1-17, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:37:y:2024:i:4:d:10.1007_s10959-024-01349-x. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.