Implicit Normalizing Flows

Cheng Lu, Jianfei Chen, Chongxuan Li, Qiuhao Wang, Jun Zhu

Keywords: Normalizing flows, deep generative models, probabilistic inference, implicit functions

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.

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One-sentence Summary: We generalize normalizing flows, allowing the mapping to be implicitly defined by the roots of an equation and enlarging the expressiveness power while retaining the tractability.

Code: [ thu-ml/implicit-normalizing-flows](https://github.com/thu-ml/implicit-normalizing-flows)

Data: [CIFAR-10](https://paperswithcode.com/dataset/cifar-10), [CIFAR-100](https://paperswithcode.com/dataset/cifar-100)

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2103.09527/code)