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# Composing Normalizing Flows for Inverse Problems

*Proceedings of the 38th International Conference on Machine Learning*, PMLR 139:11158-11169, 2021.

#### Abstract

Given an inverse problem with a normalizing flow prior, we wish to estimate the distribution of the underlying signal conditioned on the observations. We approach this problem as a task of conditional inference on the pre-trained unconditional flow model. We first establish that this is computationally hard for a large class of flow models. Motivated by this, we propose a framework for approximate inference that estimates the target conditional as a composition of two flow models. This formulation leads to a stable variational inference training procedure that avoids adversarial training. Our method is evaluated on a variety of inverse problems and is shown to produce high-quality samples with uncertainty quantification. We further demonstrate that our approach can be amortized for zero-shot inference.