# Pseudo-Invertible Neural Networks

arXiv: https://arxiv.org/abs/2602.06042 · HTML: https://mlsanity.com/paper/2602.06042/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 26 references that are in the map.
- First posted on arXiv 5 Feb 2026.

## Builds on (most cited first, 20 of 20)

- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [SDXL: Improving Latent Diffusion Models for High-Resolution Image Synthesis](https://mlsanity.com/paper/2307.01952/) (2023)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (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)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model](https://mlsanity.com/paper/2212.00490/) (2022)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [SNIPS: Solving Noisy Inverse Problems Stochastically](https://mlsanity.com/paper/2105.14951/) (2021)
- [Orthogonalizing Convolutional Layers with the Cayley Transform](https://mlsanity.com/paper/2104.07167/) (2021)
- [SurVAE Flows: Surjections to Bridge the Gap between VAEs and Flows](https://mlsanity.com/paper/2007.02731/) (2020)
- [PIE: Pseudo-Invertible Encoder](https://mlsanity.com/paper/2111.00619/) (2021)
- [Generalized Inversion of Nonlinear Operators](https://mlsanity.com/paper/2111.10755/) (2021)
- [Who Said Neural Networks Aren't Linear?](https://mlsanity.com/paper/2510.08570/) (2025)

## Built on by (most cited first, 2 of 2)

- [GSNR: Graph Smooth Null-Space Representation for Inverse Problems](https://mlsanity.com/paper/2602.20328/) (2026)
- [Moore, Escher, Penrose: A Conformal Golden Braid](https://mlsanity.com/paper/2610.02210/) (2026)

## Abstract (arXiv)

The Moore-Penrose Pseudo-inverse (PInv) serves as the fundamental solution for linear systems. In this paper, we propose a natural generalization of PInv to the nonlinear regime in general and to neural networks in particular. We introduce Surjective Pseudo-invertible Neural Networks (SPNN), a class of architectures explicitly designed to admit a tractable non-linear PInv. The proposed non-linear PInv and its implementation in SPNN satisfy fundamental geometric properties. One such property is null-space projection or "Back-Projection", x' = x + A^dagger(y-Ax), which moves a sample x to its closest consistent state x' satisfying Ax=y. We formalize Non-Linear Back-Projection (NLBP), a method that guarantees the same consistency constraint for non-linear mappings f(x)=y via our defined PInv. We leverage SPNNs to expand the scope of zero-shot inverse problems. Diffusion-based null-space projection has revolutionized zero-shot solving for linear inverse problems by exploiting closed-form back-projection. We extend this method to non-linear degradations. Here, "degradation" is broadly generalized to include any non-linear loss of information, spanning from optical distortions to semantic abstractions like classification. This approach enables zero-shot inversion of complex degradations and allows precise semantic control over generative outputs without retraining the diffusion prior.