The Riemannian Geometry of Deep Generative Models (1711.08014v1)

Abstract: Deep generative models learn a mapping from a low dimensional latent space to a high-dimensional data space. Under certain regularity conditions, these models parameterize nonlinear manifolds in the data space. In this paper, we investigate the Riemannian geometry of these generated manifolds. First, we develop efficient algorithms for computing geodesic curves, which provide an intrinsic notion of distance between points on the manifold. Second, we develop an algorithm for parallel translation of a tangent vector along a path on the manifold. We show how parallel translation can be used to generate analogies, i.e., to transport a change in one data point into a semantically similar change of another data point. Our experiments on real image data show that the manifolds learned by deep generative models, while nonlinear, are surprisingly close to zero curvature. The practical implication is that linear paths in the latent space closely approximate geodesics on the generated manifold. However, further investigation into this phenomenon is warranted, to identify if there are other architectures or datasets where curvature plays a more prominent role. We believe that exploring the Riemannian geometry of deep generative models, using the tools developed in this paper, will be an important step in understanding the high-dimensional, nonlinear spaces these models learn.

• The paper applies Riemannian geometry to study the structure of manifolds generated by deep generative models, developing algorithms for geodesic paths and parallel translation.
• It introduces efficient algorithms for computing geodesic paths and parallel translation on the learned manifold, avoiding complex second derivatives.
• Experiments show low curvature on CelebA and SVHN datasets using a VAE architecture, suggesting linear paths in latent space can approximate geodesics on the generated manifold.

Overview of The Riemannian Geometry of Deep Generative Models

The paper "The Riemannian Geometry of Deep Generative Models," by Hang Shao, Abhishek Kumar, and P. Thomas Fletcher, presents a thorough investigation into the geometric properties of manifolds generated by deep generative models. Within the framework of Riemannian geometry, the authors develop and evaluate algorithms that compute geodesic paths and enable parallel translation of tangent vectors across these manifolds. The paper emphasizes the importance of understanding nonlinear, high-dimensional spaces created by deep learning methods, which are instrumental for improving data generation quality and representation efficacy.

Core Contributions

1. Geodesic Path Computation: The authors propose an algorithm that efficiently estimates geodesic paths on the manifold induced by a deep generative model. This approach circumvents the computational burden of second derivatives and matrix inversions typically required in the continuous geodesic equation. Their techniques leverage discretized approximations to calculate geodesics, allowing interpolation between data points on the manifold with minimal alteration to the path's integrity.
2. Parallel Translation: Building on the manifold perspective, the paper introduces an algorithm for parallel translation that maintains the tangency of vectors during translation across the manifold. This algorithm enables the computation of analogies, facilitating the transfer of semantic changes from one data point to another on the manifold.
3. Empirical Findings: Experiments conducted on datasets such as CelebA and SVHN reveal that manifolds learned by the VAE architecture possess low curvature. This observation suggests a notable approximation where linear paths in the latent space closely resemble geodesics on the generated manifold. Such linear paths have practical implications for latent space traversal, resulting in plausible alterations in generated images.

Numerical Results and Observations

The paper presents several numerical results that support the hypothesis regarding the low curvature of learned manifolds. Specifically, geodesic paths exhibit shorter arc lengths compared to linear paths, albeit with less pronounced differences than initially anticipated. The paper further visualizes geodesic means and distances, establishing a slight advantage over linear techniques in grouping similar images based on attributes. However, the evidence regarding curvature's role remains modest, underscoring the need for further exploration across diverse architectures and datasets.

Implications and Future Directions

Understanding the Riemannian geometry of deep generative models bears significance for several advanced research areas, such as improving interpolation methods for latent space navigation and data generation. The insights into curvature, specifically its negligible presence, propound broader implications for the efficiency and accuracy of manifold learning techniques in unsupervised and semi-supervised learning paradigms.

Future research could span deeper investigations under varying architectures, examining whether more pronounced curvature emerges across different model configurations or data types. Furthermore, the paper's methodologies pave the way for refinement in geometric approaches, potentially enhancing interpretability and robustness in high-dimensional, non-linear model spaces.

In conclusion, by integrating geometric principles within deep generative modeling, the authors offer a compelling framework that bridges mathematical theory with computational practicality, marking a forward-thinking contribution to the understanding and development of complex model landscapes.