A friendly introduction to triangular transport

Abstract: Decision making under uncertainty is a cross-cutting challenge in science and engineering. Most approaches to this challenge employ probabilistic representations of uncertainty. In complicated systems accessible only via data or black-box models, however, these representations are rarely known. We discuss how to characterize and manipulate such representations using \textit{triangular transport maps}, which approximate any complex probability distribution as a transformation of a simple, well-understood distribution. The particular structure of triangular transport guarantees many desirable mathematical and computational properties that translate well into solving practical problems. Triangular maps are actively used for density estimation, (conditional) generative modelling, Bayesian inference, data assimilation, optimal experimental design, and related tasks. While there is ample literature on the development and theory of triangular transport methods, this manuscript provides a detailed introduction for scientists interested in employing measure transport without assuming a formal mathematical background. We build intuition for the key foundations of triangular transport, discuss many aspects of its practical implementation, and outline the frontiers of this field.

Submission Type: Long submission (more than 12 pages of main content)

Previous TMLR Submission Url: https://openreview.net/forum?id=ZhAtsyXdn4

Changes Since Last Submission: This is a corrected resubmission of a manuscript that was previously desk-rejected for a formatting issue related to line spacing. We removed the line-spacing override and regenerated the manuscript using the official TMLR style/template without modifications. We also rechecked the submission for anonymization and formatting compliance. No substantive technical changes were made.

Assigned Action Editor: ~Franck_Iutzeler1

Submission Number: 8606