Conditional Sampling and Density Estimation with Triangular Convex Flows Open Access

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

Rights statement

• Permission granted by the author to include this thesis or dissertation in this repository. All rights reserved by the author. Please contact the author for information regarding the reproduction and use of this thesis or dissertation.

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

Last modified

Primary PDF

Thumbnail	Title	Date Uploaded	Actions
	Conditional Sampling and Density Estimation with Triangular Convex Flows ()	2023-04-13 13:28:46 -0400	Press to Select an action Download

Supplemental Files

Thumbnail	Title	Date Uploaded	Actions