# Learning Identity-Preserving Transformations on Data Manifolds

arXiv: https://arxiv.org/abs/2106.12096 · HTML: https://mlsanity.com/paper/2106.12096/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 66 references that are in the map.
- First posted on arXiv 22 Jun 2021.

## Builds on (most cited first, 29 of 29)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Perceptual Losses for Real-Time Style Transfer and Super-Resolution](https://mlsanity.com/paper/1603.08155/) (2016)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [RandAugment: Practical automated data augmentation with a reduced search space](https://mlsanity.com/paper/1909.13719/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Understanding Neural Networks Through Deep Visualization](https://mlsanity.com/paper/1506.06579/) (2015)
- [Fast AutoAugment](https://mlsanity.com/paper/1905.00397/) (2019)
- [Learning with Pseudo-Ensembles](https://mlsanity.com/paper/1412.4864/) (2014)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Population Based Augmentation: Efficient Learning of Augmentation Policy Schedules](https://mlsanity.com/paper/1905.05393/) (2019)
- [The Riemannian Geometry of Deep Generative Models](https://mlsanity.com/paper/1711.08014/) (2017)
- [Metrics for Deep Generative Models](https://mlsanity.com/paper/1711.01204/) (2017)
- [Learning the Irreducible Representations of Commutative Lie Groups](https://mlsanity.com/paper/1402.4437/) (2014)
- [Variational Autoencoder with Learned Latent Structure](https://mlsanity.com/paper/2006.10597/) (2020)
- [Variational Autoencoders with Riemannian Brownian Motion Priors](https://mlsanity.com/paper/2002.05227/) (2020)
- [Fast and Robust Shortest Paths on Manifolds Learned from Data](https://mlsanity.com/paper/1901.07229/) (2019)
- [Dreaming More Data: Class-dependent Distributions over Diffeomorphisms for Learned Data Augmentation](https://mlsanity.com/paper/1510.02795/) (2015)
- [Deep Multi-task Multi-label CNN for Effective Facial Attribute Classification](https://mlsanity.com/paper/2002.03683/) (2020)
- [Disentangling Factors of Variation by Mixing Them](https://mlsanity.com/paper/1711.07410/) (2017)
- [Representing Closed Transformation Paths in Encoded Network Latent Space](https://mlsanity.com/paper/1912.02644/) (2019)
- [Exploring Explicit Domain Supervision for Latent Space Disentanglement in Unpaired Image-to-Image Translation](https://mlsanity.com/paper/1902.03782/) (2019)
- [A prior-based approximate latent Riemannian metric](https://mlsanity.com/paper/2103.05290/) (2021)

## Built on by (most cited first, 4 of 4)

- [Decomposed Linear Dynamical Systems (dLDS) for learning the latent components of neural dynamics](https://mlsanity.com/paper/2206.02972/) (2022)
- [Robust Self-Supervised Learning with Lie Groups](https://mlsanity.com/paper/2210.13356/) (2022)
- [Learning Internal Representations of 3D Transformations from 2D Projected Inputs](https://mlsanity.com/paper/2303.17776/) (2023)
- [Manifold Contrastive Learning with Variational Lie Group Operators](https://mlsanity.com/paper/2306.13544/) (2023)

## Abstract (arXiv)

Many machine learning techniques incorporate identity-preserving transformations into their models to generalize their performance to previously unseen data. These transformations are typically selected from a set of functions that are known to maintain the identity of an input when applied (e.g., rotation, translation, flipping, and scaling). However, there are many natural variations that cannot be labeled for supervision or defined through examination of the data. As suggested by the manifold hypothesis, many of these natural variations live on or near a low-dimensional, nonlinear manifold. Several techniques represent manifold variations through a set of learned Lie group operators that define directions of motion on the manifold. However, these approaches are limited because they require transformation labels when training their models and they lack a method for determining which regions of the manifold are appropriate for applying each specific operator. We address these limitations by introducing a learning strategy that does not require transformation labels and developing a method that learns the local regions where each operator is likely to be used while preserving the identity of inputs. Experiments on MNIST and Fashion MNIST highlight our model's ability to learn identity-preserving transformations on multi-class datasets. Additionally, we train on CelebA to showcase our model's ability to learn semantically meaningful transformations on complex datasets in an unsupervised manner.