On the Statistical Capacity of Deep Generative Models (2501.07763v1)

Abstract: Deep generative models are routinely used in generating samples from complex, high-dimensional distributions. Despite their apparent successes, their statistical properties are not well understood. A common assumption is that with enough training data and sufficiently large neural networks, deep generative model samples will have arbitrarily small errors in sampling from any continuous target distribution. We set up a unifying framework that debunks this belief. We demonstrate that broad classes of deep generative models, including variational autoencoders and generative adversarial networks, are not universal generators. Under the predominant case of Gaussian latent variables, these models can only generate concentrated samples that exhibit light tails. Using tools from concentration of measure and convex geometry, we give analogous results for more general log-concave and strongly log-concave latent variable distributions. We extend our results to diffusion models via a reduction argument. We use the Gromov--Levy inequality to give similar guarantees when the latent variables lie on manifolds with positive Ricci curvature. These results shed light on the limited capacity of common deep generative models to handle heavy tails. We illustrate the empirical relevance of our work with simulations and financial data.

• The paper demonstrates that DGMs using Gaussian latent variables produce light-tailed outputs, challenging the universal approximation assumption.
• It extends analysis to log-concave and strongly log-concave priors, showing consistent non-universality in sample diversity and uncertainty.
• Findings apply to diffusion models via the Gromov–Levy inequality, emphasizing the need for innovative models to capture heavy-tailed distributions.

On the Statistical Capacity of Deep Generative Models

The paper, "On the Statistical Capacity of Deep Generative Models" by Edric Tam and David B. Dunson, provides an in-depth analysis of the statistical limitations associated with deep generative models (DGMs) such as Variational Autoencoders (VAEs) and Generative Adversarial Networks (GANs). Despite their widespread application in sampling from complex and high-dimensional distributions, the assumption that DGMs can universally approximate any target distribution is critically examined and found lacking under certain conditions.

Summary of Key Findings

1. Non-Universality in DGMs:
   • The authors demonstrate that deep generative models are not universal generators as commonly assumed. Specifically, when Gaussian latent variables are employed, the generated samples are inherently light-tailed, contradicting the idea that they can approximate any continuous target distribution. This insight is primarily established using concentration of measure and convex geometry techniques.
2. Extended Analysis Beyond Gaussian Latent Variables:
   • The paper extends the analysis to log-concave and strongly log-concave latent variable distributions. Similar non-universal generation properties are identified, indicating that these distributions also result in generated samples with light-tailed properties. This generalization underscores a broader issue across latent distributions typically used in practice.
3. Diffusion Models:
   • Through a reduction argument, the authors extend their findings to include diffusion models. By leveraging the Gromov–Levy inequality, they suggest analogous limitations when latent variables lie on manifolds with positive Ricci curvature, thus highlighting the intrinsic limitations across different structural latent frameworks in DGMs.
4. Implications for Heavy-Tailed Distributions:
   • The limitations identified are particularly salient in contexts requiring heavy-tailed models, such as financial data analysis and anomaly detection. The paper illustrates that generated samples typically underestimate true distribution uncertainty and diversity when the target distribution is heavy-tailed.

Practical and Theoretical Implications

The implications of these findings are of significant import to both practitioners and theorists in AI and machine learning. Practically, the insights caution against the default use of Gaussian latent distributions in applications that are sensitive to tail properties, such as financial modeling and anomaly detection. Theoretically, this research challenges the folklore of the universal approximation capabilities of neural network-based generative models, calling for a reevaluation of the assumptions underlying their applicability.

Future Directions

Given the limitations identified, future research directions could focus on the development of novel generative frameworks or latent variables that overcome the constraints elucidated in this paper. Exploration into more sophisticated priors or alternative modeling strategies that extend beyond simple Lipschitz transformations may prove fruitful. The findings also open avenues for enhanced understanding in areas such as Bayesian inference, where posterior sampling could be advanced by accounting for heavy-tailed target distributions.

In conclusion, Tam and Dunson's work provides critical insights into the statistical capacity of deep generative models, challenging common assumptions and inviting deeper inquiry into both the theoretical foundations and practical applications of these powerful tools in machine learning.