Implicit Normalizing Flows

TL;DR

Implicit normalizing flows (ImpFlows) extend traditional flows by defining transformations implicitly, offering greater expressiveness and improved performance on density modeling and classification tasks.

Contribution

This work introduces ImpFlows, a novel class of normalizing flows that are implicitly defined, and demonstrates their theoretical advantages and empirical superiority over residual flows.

Findings

ImpFlows have a strictly richer function space than ResFlows.

ImpFlows can exactly represent functions that ResFlows approximate.

ImpFlows outperform ResFlows on multiple benchmarks with similar parameters.

Abstract

Normalizing flows define a probability distribution by an explicit invertible transformation $\boldsymbol{\mathbf{z}}=f(\boldsymbol{\mathbf{x}})$. In this work, we present implicit normalizing flows (ImpFlows), which generalize normalizing flows by allowing the mapping to be implicitly defined by the roots of an equation $F(\boldsymbol{\mathbf{z}}, \boldsymbol{\mathbf{x}})= \boldsymbol{\mathbf{0}}$. ImpFlows build on residual flows (ResFlows) with a proper balance between expressiveness and tractability. Through theoretical analysis, we show that the function space of ImpFlow is strictly richer than that of ResFlows. Furthermore, for any ResFlow with a fixed number of blocks, there exists some function that ResFlow has a non-negligible approximation error. However, the function is exactly representable by a single-block ImpFlow. We propose a scalable algorithm to train and draw samples from ImpFlows. Empirically, we evaluate ImpFlow on several classification and density modeling tasks, and ImpFlow outperforms ResFlow with a comparable amount of parameters on all the benchmarks.