A Generative Model of Symmetry Transformations (2403.01946v3)

Abstract: Correctly capturing the symmetry transformations of data can lead to efficient models with strong generalization capabilities, though methods incorporating symmetries often require prior knowledge. While recent advancements have been made in learning those symmetries directly from the dataset, most of this work has focused on the discriminative setting. In this paper, we take inspiration from group theoretic ideas to construct a generative model that explicitly aims to capture the data's approximate symmetries. This results in a model that, given a prespecified but broad set of possible symmetries, learns to what extent, if at all, those symmetries are actually present. Our model can be seen as a generative process for data augmentation. We provide a simple algorithm for learning our generative model and empirically demonstrate its ability to capture symmetries under affine and color transformations, in an interpretable way. Combining our symmetry model with standard generative models results in higher marginal test-log-likelihoods and improved data efficiency.

• The paper introduces a symmetry-aware generative model that dynamically separates latent features to capture intrinsic data symmetries.
• It employs a two-stage algorithm that first uses self-supervised learning and then applies maximum likelihood estimation for model training.
• Empirical results show that the approach boosts data efficiency and enhances test log-likelihoods by effectively incorporating symmetry transformations.

Overview of "A Generative Model of Symmetry Transformations"

The paper presents a novel approach to capturing symmetry transformations in datasets using a generative model inspired by group theory. Correctly identifying symmetries in data can leverage more efficient models with enhanced generalization capabilities. Traditional methods incorporating symmetries often require substantial prior knowledge, while recent approaches focus mainly on discriminative settings. In contrast, this work adopts a generative framework to directly learn the symmetries inherent within the data.

Key Contributions

The authors propose a Symmetry-aware Generative Model (SGM) designed to capture approximate symmetries in datasets. The model structure is devised to learn potential symmetries dynamically from data without depending on explicit prior information, thus providing a more generic solution adaptable to various datasets. The model is not only able to verify the presence of symmetries but also assesses their extents.

1. Generative Process Model: The SGM is constructed using both invariant and equivariant components. Specifically, the model separates the latent representation into components where one captures symmetry features, while the other remains invariant. It achieves this by employing a parameterized transformation $T_\eta$, which allows the model to focus on how data symmetries can be described and utilized.
2. Symmetry as Data Augmentation: The SGM conceptualizes the learning of symmetries as a process similar to data augmentation within the generative framework. By doing so, it offers interpretable learning of symmetries, such as affine and color transformations, to demonstrate their impact.
3. Two-Stage Algorithm: The model implementation is supported by a simple algorithm that first employs self-supervised learning for initial symmetry capture, followed by standard maximum likelihood estimation (MLE) for generative model training. This bifurcated strategy enhances the tractability of the model even for more complex data structures.
4. Empirical Validation: Experimental validation confirms the model's ability to learn and apply symmetries effectively, resulting in models demonstrating higher data efficiency and improved test log-likelihoods.

Methodological Insights

The proposed methodology explores learning symmetry transformations by conceptualizing a prototype-based representation. A dataset is assumed to be generated from unique prototypes that undergo symmetry transformations. This assumption challenges the model to exclusively focus on invariant characteristics within prototypical data, encouraging the automatic learning of symmetries.

The model accounts for transformations such as rotation, scaling, color shifts, and translation using parameterized representations. The learning process involves disentangling the symmetries from inherent features, thus providing a flexible framework adaptable for different datasets.

Theoretical and Practical Implications

The introduction of an SGM capable of self-discovering data symmetries has significant implications. Practically, it enables the development of models that are inherently more data-efficient, as exemplified by improved performance when integrated with standard generative models like VAEs. Theoretically, it aligns with the ongoing exploration of disentangled representation learning where isolating symmetries is crucial for comprehensive data comprehension.

Furthermore, the ability to dynamically tune symmetries broadens the applicability across diverse datasets without substantial manual intervention or pre-encoded knowledge. This flexibility could pave the way for improved model robustness and adaptability, even in scenarios characterized by limited labeled data.

Future Directions

While the paper focuses on proving the utility of the proposed model in specific settings, future work could explore broader sets of possible symmetries or delve into its application across domains like dynamic systems and structured data representation. Understanding the robustness of the SGM across varied datasets and transformations remains an exciting avenue for exploration. Additionally, leveraging invariance learning to devise more powerful data augmentation techniques in discriminative models poses a promising linkage that could further this research trajectory.

In conclusion, the paper presents a structured method for leveraging symmetry in data, inherently improving model performance through learned transformations. The approach balances theoretical elegance with practical utility, hinting at a range of applications in both academic and industry settings.