AUTM Flow: Atomic Unrestricted Time Machine for Monotonic Normalizing Flows

TL;DR

AUTM Flow introduces an integral-based method for constructing monotonic normalizing flows with explicit inverses, enabling flexible, efficient, and universal transformations for complex density modeling.

Contribution

The paper presents AUTM, a novel integral-based approach that provides explicit inverses and unrestricted parameters for monotonic normalizing flows, with proven universality and practical advantages.

Findings

AUTM flows are universal approximators of all monotonic normalizing flows.

AUTM demonstrates superior speed and memory efficiency in experiments.

The approach enables transforming existing flows into more flexible models.

Abstract

Nonlinear monotone transformations are used extensively in normalizing flows to construct invertible triangular mappings from simple distributions to complex ones. In existing literature, monotonicity is usually enforced by restricting function classes or model parameters and the inverse transformation is often approximated by root-finding algorithms as a closed-form inverse is unavailable. In this paper, we introduce a new integral-based approach termed "Atomic Unrestricted Time Machine (AUTM)", equipped with unrestricted integrands and easy-to-compute explicit inverse. AUTM offers a versatile and efficient way to the design of normalizing flows with explicit inverse and unrestricted function classes or parameters. Theoretically, we present a constructive proof that AUTM is universal: all monotonic normalizing flows can be viewed as limits of AUTM flows. We provide a concrete example to show how to approximate any given monotonic normalizing flow using AUTM flows with guaranteed convergence. The result implies that AUTM can be used to transform an existing flow into a new one equipped with explicit inverse and unrestricted parameters. The performance of the new approach is evaluated on high dimensional density estimation, variational inference and image generation. Experiments demonstrate superior speed and memory efficiency of AUTM.