On the Representation and Learning of Monotone Triangular Transport Maps

Density estimation via measure transport: Outlook for applications in the biological sciences

“…Finally, in Section 5.2.2, we discuss how, as part of the computational framework from Figure 2, measure transport methods have the potential to address other problems, in addition to the above-mentioned classification one. More precisely, by using the randomization technique we propose in Section 5.2.2 to infer the dependence structure of the underlying unknown probability measure (i.e., dependence relations between the random variables), adaptive transport maps [3] could provide a means for generating hypotheses regarding gene relationships for the radiation biology application under consideration. Before getting into such details, we now continue with an overview of required topics from measure transport theory.…”

“…To compute or, equivalently, learn the transport maps we seek, we use software [2,28] made available by the MIT Uncertainty Quantification (MUQ) Group [27]. The reader is referred to the original publications [3,10,24,39] for complete descriptions of the numerical methods and software libraries. A summary is provided next to make the present manuscript self-contained.…”

“…Many types of maps (not necessarily optimal), or couplings, exist between measures [36,45]. In this study, we consider the use of triangular transport maps, as they enable practical numerical computation of a transport map and its inverse [3,24].…”

“…Triangular maps enable efficient numerical computation of a transport map and its inverse, which is also triangular. Moreover, triangular transport maps are well-suited for conditional density estimation and sampling [3,10,17,24,39]. Techniques for computational efficiency and scalability to high-dimensional problems include, among others, composition of low order maps [10] and the use of sparse triangular transport maps [3,17,29,39].…”

“…As explained in detail in the original publications [3,24] and summarized in Appendix A of the present paper, the algorithms we use and build upon rely on finite-dimensional approximations to the functional representations for the transport map components, which may be described generally to depend on functions f i of the form…”