Conditional Sampling and Density Estimation with Triangular Convex Flows Público

Abstract

We introduce Triangular Convex Flows (TC-Flow), a method for learning conditional probability distributions given samples from the joint probability distributions. Unlike previous methods that rely on constructing monotone triangular transport maps through soft penalties and partial integration, TC-Flow uses a novel map parameterization based on the input gradient of scalar-valued fully and partially input convex neural networks (FICNN and PICNN). The approach guarantees monotone maps without requiring specific network weights and ensures optimal maps under optimal transport theory by approximating Kantorovich potentials. During training, we parallelize the process over map components and minimize the expected negative log-likelihood of the samples. We demonstrate the effectiveness of TC-Flow on synthetic two-dimensional datasets and compare it to existing triangular maps model on high-dimensional benchmark problems. Our numerical experiments show that TC-Flow is competitive with the state-of-the-art model in both expressiveness and scalability.

Table of Contents

1 Introduction ........................ 1

1.1 Contributions and Outline ........................ 3

2 Theoretical and Modeling Background ........................ 4

2.1 Optimal Transport Maps......................... 4

2.2 Monotone Triangular Transport Maps.................. 5

2.3 Maximum Likelihood Estimation .................... 6

2.4 Adaptive Transport Map Model..................... 8

3 Formulation and Analysis of Triangular Convex Flows ........................ 10

3.1 Input Convex Neural Networks ..................... 10

3.2 ICNN Triangular Maps.......................... 11

3.3 ICNN Block Triangular Maps ...................... 12

3.4 Training Problem............................. 14

3.5 Inversion Problem............................. 15

4 Numerical Experiments ........................ 17

4.1 Two Dimensional Synthetic Data .................... 17

4.2 Tabular Dataset.............................. 19

5 Discussions and Future Directions ........................ 22

6 Concluding Remarks ........................ 24

Bibliography ........................ 25 

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Applied Mathematics
Palabra Clave	Optimal Transport Triangular Transport Maps
Committee Chair / Thesis Advisor	Deepanshu Verma, Emory University
Committee Members	Lars Ruthotto, Emory University Kevin McAlister, Emory University

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