Latent Space Oddity: on the Curvature of Deep Generative Models (1710.11379v3)

Abstract: Deep generative models provide a systematic way to learn nonlinear data distributions, through a set of latent variables and a nonlinear "generator" function that maps latent points into the input space. The nonlinearity of the generator imply that the latent space gives a distorted view of the input space. Under mild conditions, we show that this distortion can be characterized by a stochastic Riemannian metric, and demonstrate that distances and interpolants are significantly improved under this metric. This in turn improves probability distributions, sampling algorithms and clustering in the latent space. Our geometric analysis further reveals that current generators provide poor variance estimates and we propose a new generator architecture with vastly improved variance estimates. Results are demonstrated on convolutional and fully connected variational autoencoders, but the formalism easily generalize to other deep generative models.

• The paper introduces a stochastic Riemannian metric that quantifies latent space curvature in deep generative models.
• The paper demonstrates that using Riemannian distances improves interpolation and clustering results compared to traditional Euclidean measures.
• The authors propose an RBF-based generator design that enhances variance estimation and captures uncertainty beyond training data regions.

Latent Space Oddity: On the Curvature of Deep Generative Models

This paper, titled "Latent Space Oddity: On the Curvature of Deep Generative Models" by Georgios Arvanitidis, Lars Kai Hansen, and Søren Hauberg, provides a systematic investigation into the geometric properties of latent spaces in deep generative models. The authors specifically focus on the curvature induced by stochastic generators, which are frequently applied in models such as Variational Autoencoders (VAEs) and Generative Adversarial Networks (GANs).

The primary contribution of this paper is the introduction and analysis of a stochastic Riemannian metric to account for the distortions in the latent space due to the nonlinearity of the generator networks. This novel geometric perspective results in substantial improvements in evaluating distances, interpolating data points, estimating probability distributions, and performing clustering operations within the latent space.

Key Contributions

1. Stochastic Riemannian Metric Characterization: The paper offers a method to describe the distortion of the latent space using a stochastic Riemannian metric. This metric accounts for the local Jacobian, which represents the change in natural distance functions, proposing that a latent space should be perceived more accurately as a curved structure rather than a flat, Euclidean space.
2. Improved Distance Measures: By leveraging this metric, the authors demonstrate that distances and interpolations between data points in the latent space reflect a more accurate geometrical and statistical structure of the data manifold. Experiments confirm that clustering using Riemannian distances results in significantly improved alignment with ground truth labels as opposed to conventional Euclidean metrics.
3. Variance Estimation Critique and Proposal: The authors identify shortcomings in current generator architectures concerning variance estimation, indicating that these models provide poor estimates beyond training data support regions. A new generator design, utilizing a radial basis function (RBF) neural network to model the inverse variance, is proposed, leading to improved variance estimation.
4. Application to Various Models: The formalism developed in the paper was applied to convolutional and fully connected VAEs but is broadly applicable to other deep generative models like GANs.

Implications and Future Directions

The implications of this meticulous geometric assessment are broad. Practically, the redefined metric can enhance generative processes in terms of efficiency and accuracy of sampling, clustering, and interpolating within latent spaces. Theoretically, this work opens new avenues for understanding how learning architectures can capitalize on manifold structures in latent space design.

Looking forward, these concepts pave the way for further exploration in leveraging geometry within neural network-based learning models, particularly focusing on incorporating observed geometric properties during model training. The proposed variance framework, which better models uncertainty and irregularities in latent structures, can prompt enhancements in other aspects of model design and performance.

Conclusion

This paper identifies and addresses a critical challenge in the understanding and application of deep generative models: the distorted view of latent spaces due to the nonlinear characteristics of underlying generators. By providing a rigorous treatment of the geometric interpretation of these spaces, aided by detailed results and practical experiments, the authors lay a robust foundation for future advancements in the field of generative modeling. The impact of better geometric and probabilistic representations has the potential to significantly influence both research innovations and practical applications in machine learning and artificial intelligence.