Symplectic Generative Networks (SGNs): A Hamiltonian Framework for Invertible Deep Generative Modeling

TL;DR

This paper introduces Symplectic Generative Networks (SGNs), a novel Hamiltonian-based deep generative model that ensures invertibility and volume preservation, enabling efficient likelihood computation and robust theoretical properties.

Contribution

The paper provides a rigorous mathematical foundation for SGNs, including proofs, complexity analysis, universal approximation, information theory, and stability, advancing the theoretical understanding of Hamiltonian generative models.

Findings

Proves invertibility and volume preservation of SGNs.

Offers a complexity analysis comparing SGNs to VAEs and normalizing flows.

Establishes universal approximation capabilities with error bounds.

Abstract

We introduce the \emph{Symplectic Generative Network (SGN)}, a deep generative model that leverages Hamiltonian mechanics to construct an invertible, volume-preserving mapping between a latent space and the data space. By endowing the latent space with a symplectic structure and modeling data generation as the time evolution of a Hamiltonian system, SGN achieves exact likelihood evaluation without incurring the computational overhead of Jacobian determinant calculations. In this work, we provide a rigorous mathematical foundation for SGNs through a comprehensive theoretical framework that includes: (i) complete proofs of invertibility and volume preservation, (ii) a formal complexity analysis with theoretical comparisons to Variational Autoencoders and Normalizing Flows, (iii) strengthened universal approximation results with quantitative error bounds, (iv) an information-theoretic analysis based on the geometry of statistical manifolds, and (v) an extensive stability analysis with adaptive integration guarantees. These contributions highlight the fundamental advantages of SGNs and establish a solid foundation for future empirical investigations and applications to complex, high-dimensional data.