# Deep Generative Models: Complexity, Dimensionality, and Approximation

arXiv: https://arxiv.org/abs/2504.00820 · HTML: https://mlsanity.com/paper/2504.00820/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 53 references that are in the map.
- First posted on arXiv 1 Apr 2025.

## Builds on (most cited first, 24 of 24)

- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [High-Resolution Image Synthesis and Semantic Manipulation with Conditional GANs](https://mlsanity.com/paper/1711.11585/) (2017)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Self-Attention Generative Adversarial Networks](https://mlsanity.com/paper/1805.08318/) (2018)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [Conditional Image Synthesis With Auxiliary Classifier GANs](https://mlsanity.com/paper/1610.09585/) (2016)
- [Towards Principled Methods for Training Generative Adversarial Networks](https://mlsanity.com/paper/1701.04862/) (2017)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Ladder Variational Autoencoders](https://mlsanity.com/paper/1602.02282/) (2016)
- [The Power of Depth for Feedforward Neural Networks](https://mlsanity.com/paper/1512.03965/) (2015)
- [HumanGAN: A Generative Model of Humans Images](https://mlsanity.com/paper/2103.06902/) (2021)
- [CVAE-GAN: Fine-Grained Image Generation through Asymmetric Training](https://mlsanity.com/paper/1703.10155/) (2017)
- [An error analysis of generative adversarial networks for learning distributions](https://mlsanity.com/paper/2105.13010/) (2021)
- [Smooth function approximation by deep neural networks with general activation functions](https://mlsanity.com/paper/1906.06903/) (2019)
- [Statistical guarantees for generative models without domination](https://mlsanity.com/paper/2010.09237/) (2020)
- [Intrinsic Dimension Estimation Using Wasserstein Distances](https://mlsanity.com/paper/2106.04018/) (2021)
- [Learning Manifold Dimensions with Conditional Variational Autoencoders](https://mlsanity.com/paper/2302.11756/) (2023)
- [On the capacity of deep generative networks for approximating distributions](https://mlsanity.com/paper/2101.12353/) (2021)
- [Wasserstein Variational Inference](https://mlsanity.com/paper/1805.11284/) (2018)
- [Generalization Properties of Optimal Transport GANs with Latent Distribution Learning](https://mlsanity.com/paper/2007.14641/) (2020)
- [On Deep Generative Models for Approximation and Estimation of Distributions on Manifolds](https://mlsanity.com/paper/2302.13183/) (2023)
- [A likelihood approach to nonparametric estimation of a singular distribution using deep generative models](https://mlsanity.com/paper/2105.04046/) (2021)

## Built on by (most cited first, 4 of 4)

- [Off-policy Reinforcement Learning with Model-based Exploration Augmentation](https://mlsanity.com/paper/2510.25529/) (2025)
- [A Cycle-Consistency Constrained Framework for Dynamic Solution Space Reduction in Noninjective Regression](https://mlsanity.com/paper/2507.04659/) (2025)
- [On the Effect of Regularization on Nonparametric Mean-Variance Regression](https://mlsanity.com/paper/2511.22004/) (2025)
- [Identifiable Deep Latent Variable Models for MNAR Data](https://mlsanity.com/paper/2603.24771/) (2026)

## Abstract (arXiv)

Generative networks have shown remarkable success in learning complex data distributions, particularly in generating high-dimensional data from lower-dimensional inputs. While this capability is well-documented empirically, its theoretical underpinning remains unclear. One common theoretical explanation appeals to the widely accepted manifold hypothesis, which suggests that many real-world datasets, such as images and signals, often possess intrinsic low-dimensional geometric structures. Under this manifold hypothesis, it is widely believed that to approximate a distribution on a d-dimensional Riemannian manifold, the latent dimension needs to be at least d or d+1. In this work, we show that this requirement on the latent dimension is not necessary by demonstrating that generative networks can approximate distributions on d-dimensional Riemannian manifolds from inputs of any arbitrary dimension, even lower than d, taking inspiration from the concept of space-filling curves. This approach, in turn, leads to a super-exponential complexity bound of the deep neural networks through expanded neurons. Our findings thus challenge the conventional belief on the relationship between input dimensionality and the ability of generative networks to model data distributions. This novel insight not only corroborates the practical effectiveness of generative networks in handling complex data structures, but also underscores a critical trade-off between approximation error, dimensionality, and model complexity.