Year: 2025
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Wang, Kevin, Niu, Hongqian, Wang, Yixin, Li, Didong
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
Awards: Toward Automated Uncertainty Quantification in Causal Inference, ACED: Tail-aware Generative Modeling for Inverse Discovery of Molecules, Collaborative Research: ATD: Hawkes Process-Based Causal Relationship Discovery For Complex Threat Detection and Forecasting, CAREER: Bringing Structure to the Unstructured: Robust Causal and Statistical Modeling of High-dimensional Unstructured Data, EAGER: ADAPT: Hypotheses Generation in Heterogeneous Catalysis using Causal Inference and Machine Learning
OpenAlex ID: W4417228673
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