Unsupervised Learning of Representations Invariant and Equivariant to Group Transformations

מושגי ליבה

This work introduces a novel framework for unsupervised learning of representations that are separated into a group-invariant component and a group-equivariant component, valid for any group G.

תקציר

The key contributions of this work are:

1. Introduction of a novel framework for learning group equivariant representations, where the latent representation is separated into an invariant and an equivariant component.

2. Characterization of the mathematical conditions for the group action function component and proposal of an explicit construction suitable for any group G. This is the first method for unsupervised learning of separated invariant-equivariant representations valid for any group.

3. Experimental validation of the framework on diverse data types (MNIST, sets of digits, point clouds, molecular conformations) and different network architectures, demonstrating the flexibility and validity of the approach.

The proposed framework learns to encode data into a group-invariant latent code and a group action. By separating the embedding into an invariant and an equivariant part, the method can learn expressive low-dimensional group-invariant representations using the power of autoencoders.

The key idea is that the network learns to encode and decode data to and from a group-invariant representation by additionally learning to predict the appropriate group action to align input and output. The authors derive the necessary conditions on the equivariant encoder and present a construction valid for any group G, both discrete and continuous.

The experiments demonstrate the effectiveness of the approach. For example, on rotated MNIST, the model can reconstruct rotated versions of digits by predicting the appropriate rotation. On sets of digits, the model can compress the set information into a much lower-dimensional representation compared to a non-invariant autoencoder. Similarly, on point cloud and molecular conformation data, the model learns representations that are invariant to translations, rotations and permutations.

סטטיסטיקה

The data used in the experiments includes:

Rotated MNIST dataset
Sets of digits with varying sizes
Tetris shape point cloud dataset
QM9 molecular conformation dataset
ShapeNet 3D object dataset

ציטוטים

"Equivariant neural networks, whose hidden features transform according to representations of a group G acting on the data, exhibit training efficiency and an improved generalisation performance."
"We introduce a novel framework for unsupervised learning of representations that are separated into a group-invariant component and a group-equivariant component, valid for any group G."
"Our representations are by construction separated in an invariant and equivariant component."

תובנות מפתח מזוקקות מ:

Unsupervised Learning of Group Invariant and Equivariant Representations

by Robi... ב- arxiv.org 04-15-2024

https://arxiv.org/pdf/2202.07559.pdf

שאלות מעמיקות

How could this framework be extended to learn representations that are disentangled into multiple group actions, rather than just a single group action

To extend the framework to learn representations disentangled into multiple group actions, we can modify the architecture to include separate branches for each group action. Each branch would predict a different group action, allowing the model to capture the different transformations present in the data. By training the model to predict multiple group actions simultaneously, we can disentangle the representations into distinct components corresponding to each group action. This approach would require careful design of the network to ensure that the different branches can effectively learn and predict the various group transformations present in the data.

How might this approach perform on more complex, real-world datasets with hierarchical or compositional group structures

The proposed framework could perform well on more complex, real-world datasets with hierarchical or compositional group structures by adapting the network architecture and training procedure. For datasets with hierarchical group structures, the model can be designed to capture the different levels of hierarchy by incorporating multiple layers of group transformations. This would allow the model to learn representations that encode the hierarchical relationships within the data. Additionally, for datasets with compositional group structures, the framework can be extended to handle combinations of different group actions by incorporating multiple group function components. By training the model on diverse datasets with complex group structures, the framework can learn to extract meaningful representations that capture the underlying symmetries effectively.

Could the group action prediction component be further improved by incorporating uncertainty or probabilistic modeling, to better capture ambiguity or multimodality in the group transformations

Incorporating uncertainty or probabilistic modeling into the group action prediction component can enhance the model's ability to capture ambiguity or multimodality in the group transformations. By introducing probabilistic modeling, the model can learn a distribution over possible group actions rather than predicting a single deterministic transformation. This can help the model account for uncertainty in the group transformations present in the data and provide a more robust representation learning framework. By incorporating uncertainty estimation techniques such as Bayesian neural networks or variational inference, the model can capture the inherent uncertainty in the group action predictions and provide more reliable representations for ambiguous or multimodal group transformations.