Explorations in Homeomorphic Variational Auto-Encoding (1807.04689v1)

Abstract: The manifold hypothesis states that many kinds of high-dimensional data are concentrated near a low-dimensional manifold. If the topology of this data manifold is non-trivial, a continuous encoder network cannot embed it in a one-to-one manner without creating holes of low density in the latent space. This is at odds with the Gaussian prior assumption typically made in Variational Auto-Encoders (VAEs), because the density of a Gaussian concentrates near a blob-like manifold. In this paper we investigate the use of manifold-valued latent variables. Specifically, we focus on the important case of continuously differentiable symmetry groups (Lie groups), such as the group of 3D rotations $\operatorname{SO}(3)$. We show how a VAE with $\operatorname{SO}(3)$-valued latent variables can be constructed, by extending the reparameterization trick to compact connected Lie groups. Our experiments show that choosing manifold-valued latent variables that match the topology of the latent data manifold, is crucial to preserve the topological structure and learn a well-behaved latent space.

A Study on Homeomorphic Variational Auto-Encoding

The paper "Explorations in Homeomorphic Variational Auto-Encoding" investigates the intrinsic challenges posed by non-trivially topological data manifolds in the implementation of Variational Auto-Encoders (VAEs). This research focuses on leveraging manifold-valued latent variables, particularly within the scope of continuously differentiable symmetry groups, to align the topology of latent spaces more closely with the manifolds from which data are sampled.

Summary of Contributions

The authors tackle the mismatch between high-dimensional manifolds and blob-like latent spaces imposed by Gaussian assumptions in traditional VAEs. The primary innovation in this work involves the construction of VAEs using latent variables that reside on Lie groups, particularly focusing on the group of 3D rotations, $\SO3$. This is achieved through the following contributions:

1. Generalizing the Reparameterization Trick for Lie Groups: The authors extend the reparameterization trick central to VAEs to be applicable to compact connected Lie groups, allowing for manifold-compatible latent representations.
2. Preserving Topological Structures: The paper outlines the methodological details on how to construct encoders that maintain homeomorphic mappings between data and latent spaces. This focuses on learning structures that faithfully represent underlying manifold structures.
3. Decoders Utilizing Group Actions: The research introduces a novel decoder design that leverages group actions to ensure that the latent space respects and preserves the group symmetries characteristic of the data.

Methodological Insights

At the core of this paper is the utilization of the $\SO3$ group, which embodies 3D rotations, and typically presents as a non-trivial topological space that cannot be stitched onto typical blob-like Gaussian priors without introducing significant representational artifacts. The authors propose a reparameterization trick for $\SO3$, utilizing concepts from Lie groups and algebras, and proving that such parametrizations are absolutely continuous with respect to the natural Haar measure, thus ensuring well-defined densities.

Additionally, encoder networks are structured to respect the manifold topology, achieved by building surjective functions that map higher-dimensional input spaces to manifold-conforming latent spaces. This methodology enables topological alignment between data and latent spaces without the imposition of discontinuities that typically plague naive embedding strategies.

Empirical Evaluations

The empirical evaluation conducted by the authors utilizes both synthetic data constructions of $\SO3$ submanifolds within high-dimensional embeddings and real-world scenarios such as rendered 3D rotations of geometric objects. These experiments reveal that VAEs with appropriately structured latent manifolds vastly outperform those relying on traditional Gaussian spaces, particularly in maintaining continuity and coherence across interpolations of the latent space.

Implications and Future Directions

The implications of this research are profound for fields such as computer vision and robotics, where understanding and encoding non-linear transformations like rotations is pivotal. By aligning VAEs with the topological and geometric properties of the underlying data manifolds, this work opens the door to more expressive and interpretable generative models.

Future work can expand upon this foundation by exploring its applicability to other complex manifolds in real-world data and extending these principles to larger classes of Lie groups, such as $\operatorname{SE}(3)$, which incorporates both rotations and translations. Broader applicability can further enhance diverse machine learning applications, including but not limited to, learning disentangled representations and unsupervised pose estimation.

In conclusion, this paper provides a substantive advance in constructing generative models that are both theoretically sound and practically applicable, harnessing the power of manifold symmetries to address fundamental challenges in representation learning.