Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections (2404.02954v2)

Abstract: In recent years there has been increased interest in understanding the interplay between deep generative models (DGMs) and the manifold hypothesis. Research in this area focuses on understanding the reasons why commonly-used DGMs succeed or fail at learning distributions supported on unknown low-dimensional manifolds, as well as developing new models explicitly designed to account for manifold-supported data. This manifold lens provides both clarity as to why some DGMs (e.g. diffusion models and some generative adversarial networks) empirically surpass others (e.g. likelihood-based models such as variational autoencoders, normalizing flows, or energy-based models) at sample generation, and guidance for devising more performant DGMs. We carry out the first survey of DGMs viewed through this lens, making two novel contributions along the way. First, we formally establish that numerical instability of likelihoods in high ambient dimensions is unavoidable when modelling data with low intrinsic dimension. We then show that DGMs on learned representations of autoencoders can be interpreted as approximately minimizing Wasserstein distance: this result, which applies to latent diffusion models, helps justify their outstanding empirical results. The manifold lens provides a rich perspective from which to understand DGMs, and we aim to make this perspective more accessible and widespread.

Deep Generative Models: A Manifold Perspective

Generative models are a cornerstone of contemporary machine learning, taking center stage in numerous applications ranging from synthetic image generation to novel drug discovery. Alongside this proliferation of use cases, there's been a growing interest in understanding the theoretical underpinnings of Deep Generative Models (DGMs), notably through the manifold hypothesis. The essence of this hypothesis posits that high-dimensional data (e.g., images) tend to concentrate around low-dimensional subspaces or manifolds within their ambient space. This perspective not only offers theoretical clarity but also guides the development of more effective generative models. This blog post explores the field of DGMs through the manifold hypothesis, exploring both theoretical insights and practical implications drawn from "Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections".

Manifold Hypothesis and DGMs

The manifold hypothesis is instrumental in elucidating why certain DGMs, particularly diffusion models and latent variants, exhibit superior performance over others. By asserting that data lies on an unknown low-dimensional manifold within a high-dimensional space, the hypothesis highlights that successful DGMs are those capable of learning these manifold structures. This insight not only sheds light on the empirical success of specific models but also directs the development of new, more efficient algorithms.

Numerical Instability in High-Dimensional Likelihoods

A key contribution of this survey is the formal establishment of the numerical instability inherent in high-dimensional likelihood-based models when attempting to model data residing in low-dimensional spaces. This phenomenon, dubbed "manifold overfitting", arises from the models' propensity to assign disproportionate likelihoods to data points, leading to numerical instability. Importantly, this instability is shown to be unavoidable, signaling a cautionary note for the development of likelihood-based DGMs under the manifold hypothesis.

Two-Step Models and Wasserstein Distance

The survey introduces an intriguing perspective on two-step models, proposing that such models can be conceptualized as minimizing an upper bound of the Wasserstein distance, a form of optimal transport cost, between the model and the data distribution. This bound is shown to tighten at optimality under conditions of perfect data reconstruction, offering a novel interpretation of two-step models' objectives and emphasizing the utility of Wasserstein distance in developing manifold-aware DGMs.

Practical Implications and Future Directions

Understanding DGMs through the manifold hypothesis not only enhances theoretical comprehension but also has significant practical implications. For instance, it underlines the necessity of designing models that are explicitly or implicitly aware of the data's manifold structure to prevent manifold overfitting and ensure numerical stability. Moreover, the connection between two-step models and Wasserstein distance minimization opens new avenues for creating more robust and effective generative models by closely aligning their objectives with the geometry of the data manifold.

In conclusion, the survey provides a comprehensive overview of DGMs from the perspective of the manifold hypothesis, offering both novel insights and reinforcing established theories. By elucidating the challenges of numerical instability and proposing innovative interpretations of two-step models, it lays the groundwork for future research aimed at harnessing the full potential of DGMs in learning complex data distributions. As the field advances, integrating these manifold-aware methodologies will undoubtedly be crucial in unlocking new capabilities and applications for deep generative models.

Acknowledgments

This summary discusses the paper "Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections", authored by Gabriel Loaiza-Ganem, Brendan Leigh Ross, Rasa Hosseinzadeh, Anthony L. Caterini, and Jesse C. Cresswell, highlighting its key contributions and implications in the field of machine learning and deep generative models.