# Transformation Properties of Learned Visual Representations

arXiv: https://arxiv.org/abs/1412.7659 · HTML: https://mlsanity.com/paper/1412.7659/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Dec 2014.

## Built on by (most cited first, 18 of 18)

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- [Enriching Disentanglement: From Logical Definitions to Quantitative Metrics](https://mlsanity.com/paper/2305.11512/) (2023)
- [Unsupervised Representation Learning from Sparse Transformation Analysis](https://mlsanity.com/paper/2410.05564/) (2024)
- [Stimulus symmetries can confound representational similarity analyses](https://mlsanity.com/paper/2605.21324/) (2026)
- [Topographic VAEs learn Equivariant Capsules](https://mlsanity.com/paper/2109.01394/) (2021)
- [Transformation Coding: Simple Objectives for Equivariant Representations](https://mlsanity.com/paper/2202.10930/) (2022)
- [Warped Convolutional Networks: Bridge Homography to sl(3) algebra by Group Convolution](https://mlsanity.com/paper/2206.11657/) (2022)
- [Zero-Shot Transfer in Imitation Learning](https://mlsanity.com/paper/2310.06710/) (2023)
- [Algebraic Priors for Approximately Equivariant Networks](https://mlsanity.com/paper/2506.08244/) (2025)
- [Equivariant Latent Alignment via Flow Matching under Group Symmetries](https://mlsanity.com/paper/2605.30705/) (2026)

## Abstract (arXiv)

When a three-dimensional object moves relative to an observer, a change occurs on the observer's image plane and in the visual representation computed by a learned model. Starting with the idea that a good visual representation is one that transforms linearly under scene motions, we show, using the theory of group representations, that any such representation is equivalent to a combination of the elementary irreducible representations. We derive a striking relationship between irreducibility and the statistical dependency structure of the representation, by showing that under restricted conditions, irreducible representations are decorrelated. Under partial observability, as induced by the perspective projection of a scene onto the image plane, the motion group does not have a linear action on the space of images, so that it becomes necessary to perform inference over a latent representation that does transform linearly. This idea is demonstrated in a model of rotating NORB objects that employs a latent representation of the non-commutative 3D rotation group SO(3).