# Manifold Modeling in Quotient Space: Learning An Invariant Mapping with Decodability of Image Patches

arXiv: https://arxiv.org/abs/2203.05134 · HTML: https://mlsanity.com/paper/2203.05134/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 46 references that are in the map.
- First posted on arXiv 10 Mar 2022.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Non-local Neural Networks](https://mlsanity.com/paper/1711.07971/) (2017)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Real-ESRGAN: Training Real-World Blind Super-Resolution with Pure Synthetic Data](https://mlsanity.com/paper/2107.10833/) (2021)
- [Self-Attention Generative Adversarial Networks](https://mlsanity.com/paper/1805.08318/) (2018)
- [Learning Representations by Maximizing Mutual Information Across Views](https://mlsanity.com/paper/1906.00910/) (2019)
- [Noise2Noise: Learning Image Restoration without Clean Data](https://mlsanity.com/paper/1803.04189/) (2018)
- [Noise2Void - Learning Denoising from Single Noisy Images](https://mlsanity.com/paper/1811.10980/) (2018)
- [SinGAN: Learning a Generative Model from a Single Natural Image](https://mlsanity.com/paper/1905.01164/) (2019)
- [Discriminative Unsupervised Feature Learning with Exemplar Convolutional Neural Networks](https://mlsanity.com/paper/1406.6909/) (2014)
- ["Zero-Shot" Super-Resolution using Deep Internal Learning](https://mlsanity.com/paper/1712.06087/) (2017)
- [Noise2Self: Blind Denoising by Self-Supervision](https://mlsanity.com/paper/1901.11365/) (2019)
- [Deep Decoder: Concise Image Representations from Untrained Non-convolutional Networks](https://mlsanity.com/paper/1810.03982/) (2018)
- [Multi-channel Weighted Nuclear Norm Minimization for Real Color Image Denoising](https://mlsanity.com/paper/1705.09912/) (2017)
- [Gromov-Wasserstein Averaging in a Riemannian Framework](https://mlsanity.com/paper/1910.04308/) (2019)
- [Representations, Metrics and Statistics For Shape Analysis of Elastic Graphs](https://mlsanity.com/paper/2003.00287/) (2020)
- [Generative Patch Priors for Practical Compressive Image Recovery](https://mlsanity.com/paper/2006.10873/) (2020)

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

- [Self-Supervised Detection of Perfect and Partial Input-Dependent Symmetries](https://mlsanity.com/paper/2312.12223/) (2023)
- [RECON: Robust symmetry discovery via Explicit Canonical Orientation Normalization](https://mlsanity.com/paper/2505.13289/) (2025)

## Abstract (arXiv)

This study proposes a framework for manifold learning of image patches using the concept of equivalence classes: manifold modeling in quotient space (MMQS). In MMQS, we do not consider a set of local patches of the image as it is, but rather the set of their canonical patches obtained by introducing the concept of equivalence classes and performing manifold learning on their canonical patches. Canonical patches represent equivalence classes, and their auto-encoder constructs a manifold in the quotient space. Based on this framework, we produce a novel manifold-based image model by introducing rotation-flip-equivalence relations. In addition, we formulate an image reconstruction problem by fitting the proposed image model to a corrupted observed image and derive an algorithm to solve it. Our experiments show that the proposed image model is effective for various self-supervised image reconstruction tasks, such as image inpainting, deblurring, super-resolution, and denoising.