Disentangled Representation Learning with the Gromov-Monge Gap (2407.07829v2)

Abstract: Learning disentangled representations from unlabelled data is a fundamental challenge in machine learning. Solving it may unlock other problems, such as generalization, interpretability, or fairness. Although remarkably challenging to solve in theory, disentanglement is often achieved in practice through prior matching. Furthermore, recent works have shown that prior matching approaches can be enhanced by leveraging geometrical considerations, e.g., by learning representations that preserve geometric features of the data, such as distances or angles between points. However, matching the prior while preserving geometric features is challenging, as a mapping that fully preserves these features while aligning the data distribution with the prior does not exist in general. To address these challenges, we introduce a novel approach to disentangled representation learning based on quadratic optimal transport. We formulate the problem using Gromov-Monge maps that transport one distribution onto another with minimal distortion of predefined geometric features, preserving them as much as can be achieved. To compute such maps, we propose the Gromov-Monge-Gap (GMG), a regularizer quantifying whether a map moves a reference distribution with minimal geometry distortion. We demonstrate the effectiveness of our approach for disentanglement across four standard benchmarks, outperforming other methods leveraging geometric considerations.

• The paper introduces a novel framework for disentangled representation learning that uses the Gromov-Monge Gap (GMG) as a regularizer to preserve geometric properties during transformation.
• Empirical results show that employing the GMG consistently enhances disentanglement performance across various benchmark datasets, particularly within conformal regularization settings.
• The study demonstrates that GMG enables decoder-free disentanglement, offering a method to learn meaningful representations without relying on reconstruction losses, suggesting pathways for more scalable unsupervised learning.

An Overview of Disentangled Representation Learning through Geometry Preservation with the Gromov-Monge Gap

The paper "Disentangled Representation Learning through Geometry Preservation with the Gromov-Monge Gap" introduces a novel methodology for disentangled representation learning using a geometric lens, leveraging concepts from optimal transport (OT) theory. Disentangled representation learning remains a significant challenge in machine learning, where the fundamental aim is to unlock insightful, low-dimensional representations from high-dimensional data. The authors address the issue by highlighting the potential of geometry-preserving transformations, following recent studies that demonstrate the importance of local isometry and non-Gaussianity in enabling disentanglement.

Theoretical Framework and Contributions

The authors propose a framework based on the Gromov-Monge problem, a variation of the optimal transport problem geared towards isometric mappings between distributions supported on different spaces. To this end, they introduce the Gromov-Monge Gap (GMG), a regularizer designed to quantify the degree of geometry preservation by a transformation between two distributions, emphasizing the extent to which scaled distances and angles are retained. The GMG thus functions as a debiased distortion measure by comparing a given map's distortion to the minimal possible distortion.

A detailed theoretical analysis regards GMG as weakly convex, establishing more favorable optimization properties compared to classical distortion measures. Theoretical results include an examination of how the GMG behaves concerning reference distributions and the demonstration of its weak convexity across specific cost choices and setups.

Empirical Insights

Empirical results substantiate the theoretical framework, revealing that employing the GMG enhances disentanglement efficacy across multiple standard benchmarks, especially when compared to distortion alone. The authors perform extensive evaluations on well-known datasets such as Shapes3D, DSprites, SmallNORB, and Cars3D. They observe consistent improvements in disentanglement metrics when implementing the GMG, especially within conformal regularization settings, which emphasizes angle preservation. These findings reflect the GMG's potential to serve as a flexible and effective disentanglement regularizer across a spectrum of models and tasks.

Decoder-Free Disentanglement

An innovative aspect of the paper is its exploration of decoder-free disentangled representation learning. Typically, VAE architectures hinge on reconstruction losses that depend on decoder networks. The paper demonstrates how GMG enables disentanglement without this reliance, effectively learning meaningful representations in settings devoid of reconstructions, suggesting pathways towards more scalable unsupervised learning.

Future Directions

This work opens several opportunities for future research. The adaptability of the GMG further encourages exploration in self-supervised and weakly supervised scenarios, bridging gaps between disparate representation learning paradigms. Moreover, its scalability suggests potential applicability in large-scale environments where computational constraints are predominant. The intriguing prospect of encoder-only settings further indicates pathways to integrate disentanglement principles within broader lines of ongoing AI research such as fairness, interpretability, and robustness.

In conclusion, by strategically infusing geometric principles into the fabric of representation learning, the paper significantly contributes to the ongoing discourse surrounding disentangled representations, pointing to a future where machine learning models can more robustly extract and utilize the intrinsic variational factors within data.