# Trumpets: Injective Flows for Inference and Inverse Problems

arXiv: https://arxiv.org/abs/2102.10461 · HTML: https://mlsanity.com/paper/2102.10461/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 39 references that are in the map.
- First posted on arXiv 20 Feb 2021.

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Towards Principled Methods for Training Generative Adversarial Networks](https://mlsanity.com/paper/1701.04862/) (2017)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [i-RevNet: Deep Invertible Networks](https://mlsanity.com/paper/1802.07088/) (2018)
- [Flows for simultaneous manifold learning and density estimation](https://mlsanity.com/paper/2003.13913/) (2020)
- [Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators](https://mlsanity.com/paper/2006.11469/) (2020)
- [Solving Linear Inverse Problems Using GAN Priors: An Algorithm with Provable Guarantees](https://mlsanity.com/paper/1802.08406/) (2018)
- [Deep Probabilistic Imaging: Uncertainty Quantification and Multi-modal Solution Characterization for Computational Imaging](https://mlsanity.com/paper/2010.14462/) (2020)
- [Regularized Autoencoders via Relaxed Injective Probability Flow](https://mlsanity.com/paper/2002.08927/) (2020)
- [Globally Injective ReLU Networks](https://mlsanity.com/paper/2006.08464/) (2020)
- [Normalizing Flows Across Dimensions](https://mlsanity.com/paper/2006.13070/) (2020)

## Built on by (most cited first, 30 of 36)

- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Tractable Density Estimation on Learned Manifolds with Conformal Embedding Flows](https://mlsanity.com/paper/2106.05275/) (2021)
- [Diagnosing and Fixing Manifold Overfitting in Deep Generative Models](https://mlsanity.com/paper/2204.07172/) (2022)
- [Reliable amortized variational inference with physics-based latent distribution correction](https://mlsanity.com/paper/2207.11640/) (2022)
- [Generalized Normalizing Flows via Markov Chains](https://mlsanity.com/paper/2111.12506/) (2021)
- [Lifting Architectural Constraints of Injective Flows](https://mlsanity.com/paper/2306.01843/) (2023)
- [Manifold Learning by Mixture Models of VAEs for Inverse Problems](https://mlsanity.com/paper/2303.15244/) (2023)
- [Universal Joint Approximation of Manifolds and Densities by Simple Injective Flows](https://mlsanity.com/paper/2110.04227/) (2021)
- [Conditional Injective Flows for Bayesian Imaging](https://mlsanity.com/paper/2204.07664/) (2022)
- [Deep Bayesian inference for seismic imaging with tasks](https://mlsanity.com/paper/2110.04825/) (2021)
- [VQ-Flows: Vector Quantized Local Normalizing Flows](https://mlsanity.com/paper/2203.11556/) (2022)
- [Injectivity of ReLU networks: perspectives from statistical physics](https://mlsanity.com/paper/2302.14112/) (2023)
- [Image Reconstruction without Explicit Priors](https://mlsanity.com/paper/2303.12217/) (2023)
- [Nonlinear Isometric Manifold Learning for Injective Normalizing Flows](https://mlsanity.com/paper/2203.03934/) (2022)
- [Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spaces](https://mlsanity.com/paper/2205.14627/) (2022)
- [Denoising Deep Generative Models](https://mlsanity.com/paper/2212.01265/) (2022)
- [Neural Implicit Manifold Learning for Topology-Aware Density Estimation](https://mlsanity.com/paper/2206.11267/) (2022)
- [Discovering Structure From Corruption for Unsupervised Image Reconstruction](https://mlsanity.com/paper/2304.05589/) (2023)
- [Principal Manifold Flows](https://mlsanity.com/paper/2202.07037/) (2022)
- [FunkNN: Neural Interpolation for Functional Generation](https://mlsanity.com/paper/2212.14042/) (2022)
- [Deep Injective Prior for Inverse Scattering](https://mlsanity.com/paper/2301.03092/) (2023)
- [An Integral Projection-based Semantic Autoencoder for Zero-Shot Learning](https://mlsanity.com/paper/2306.14628/) (2023)
- [Accelerating Inverse Learning via Intelligent Localization with Exploratory Sampling](https://mlsanity.com/paper/2212.01016/) (2022)
- [Deep Variational Inverse Scattering](https://mlsanity.com/paper/2212.04309/) (2022)
- [Analyzing Generative Models by Manifold Entropic Metrics](https://mlsanity.com/paper/2410.19426/) (2024)
- [LoFi: Neural Local Fields for Scalable Image Reconstruction](https://mlsanity.com/paper/2411.04995/) (2024)
- [X2CT-FLOW: Maximum a posteriori reconstruction using a progressive flow-based deep generative model for ultra sparse-view computed tomography in ultra low-dose protocols](https://mlsanity.com/paper/2104.04179/) (2021)
- [Differentiable Uncalibrated Imaging](https://mlsanity.com/paper/2211.10525/) (2022)
- [Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems: An Overview](https://mlsanity.com/paper/2402.12072/) (2024)

## Abstract (arXiv)

We propose injective generative models called Trumpets that generalize invertible normalizing flows. The proposed generators progressively increase dimension from a low-dimensional latent space. We demonstrate that Trumpets can be trained orders of magnitudes faster than standard flows while yielding samples of comparable or better quality. They retain many of the advantages of the standard flows such as training based on maximum likelihood and a fast, exact inverse of the generator. Since Trumpets are injective and have fast inverses, they can be effectively used for downstream Bayesian inference. To wit, we use Trumpet priors for maximum a posteriori estimation in the context of image reconstruction from compressive measurements, outperforming competitive baselines in terms of reconstruction quality and speed. We then propose an efficient method for posterior characterization and uncertainty quantification with Trumpets by taking advantage of the low-dimensional latent space.