AUTM Flow: Atomic Unrestricted Time Machine for Monotonic Normalizing Flows
This work addresses a bottleneck in normalizing flows for machine learning practitioners by enabling more efficient and versatile flow designs, though it is incremental as it builds on existing monotonic flow methods.
The paper tackles the problem of constructing invertible triangular mappings in normalizing flows by introducing AUTM Flow, which provides explicit inverse transformations and unrestricted parameters, resulting in superior speed and memory efficiency in experiments on density estimation, variational inference, and image generation.
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.