## Implicit Normalizing Flows

Sep 28, 2020 (edited Mar 17, 2021)ICLR 2021 SpotlightReaders: Everyone
• 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.
• Supplementary Material: zip
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