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

Agnideep Aich, Ashit Aich

TL;DR

Symplectic Generative Networks (SGNs) introduce a Hamiltonian-based, volume-preserving latent transport to enable exact likelihoods without Jacobian-determinant computations. By imposing a canonical symplectic structure on the latent space and discretizing dynamics with a stable leapfrog integrator, SGNs achieve ${|\det D\Phi_T(z)|=1}$, allowing an exact likelihood in SGN-Flow and a simplified ELBO in SGN-VAE. The work provides formal complexity advantages ${\mathcal{O}(T\cdot d)}$ over traditional normalizing flows, strengthened universal-approximation results for volume-preserving maps, and an information-geometric interpretation linking geodesics on statistical manifolds to SGN dynamics, along with a rigorous stability and backward-error analysis. It also extends to non-volume-preserving maps via a density-correction composition, and outlines a practical training algorithm, highlighting potential applications in physics-informed modeling and high-dimensional data generation. These results establish SGNs as a principled, efficient alternative to classical invertible models with strong theoretical guarantees and broad applicability.

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.

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

TL;DR

Symplectic Generative Networks (SGNs) introduce a Hamiltonian-based, volume-preserving latent transport to enable exact likelihoods without Jacobian-determinant computations. By imposing a canonical symplectic structure on the latent space and discretizing dynamics with a stable leapfrog integrator, SGNs achieve , allowing an exact likelihood in SGN-Flow and a simplified ELBO in SGN-VAE. The work provides formal complexity advantages over traditional normalizing flows, strengthened universal-approximation results for volume-preserving maps, and an information-geometric interpretation linking geodesics on statistical manifolds to SGN dynamics, along with a rigorous stability and backward-error analysis. It also extends to non-volume-preserving maps via a density-correction composition, and outlines a practical training algorithm, highlighting potential applications in physics-informed modeling and high-dimensional data generation. These results establish SGNs as a principled, efficient alternative to classical invertible models with strong theoretical guarantees and broad applicability.

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.

Paper Structure

This paper contains 36 sections, 24 theorems, 141 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. SGN-Flow vs. Invertible Likelihood Models
  4. SGN-VAE vs. VAEs with Latent Dynamics
  5. Physics-Informed and Structured Generative Models
  6. Symplectic Generative Networks (SGNs)
  7. Phase Space, Prior, and Hamiltonian Dynamics
  8. Generative viewpoint.
  9. Symplectic Time Discretization
  10. Likelihoods and Training Objectives
  11. (A) SGN-Flow (exact log-likelihood).
  12. (B) SGN-VAE (ELBO).
  13. Regularity and Design Assumptions
  14. Practical Parameterizations
  15. Algorithmic Sketch
  16. ...and 21 more sections

Key Result

Theorem 4.1

Let $\Phi_T:\mathbb{R}^{2d}\to\mathbb{R}^{2d}$ be the flow obtained by integrating eq:hamilton using a symplectic integrator with step size $\Delta t$ over $T$ steps. Then, $\Phi_T$ is invertible and volume preserving:

Figures (4)

  • Figure 1: Hamiltonian dynamics in phase space. Concentric blue curves are constant energy; the red curve shows the flow.
  • Figure 2: Leapfrog integration used in SGNs.
  • Figure 3: SGN pipeline: encoder $\to$ symplectic flow $\to$ decoder. The flow is volume-preserving, so only the terminal mapping contributes a Jacobian term.
  • Figure 4: Information Geometry on Statistical Manifolds. The ellipse represents a statistical manifold endowed with the Fisher-Rao metric, while the two curves illustrate a geodesic and a Hamiltonian flow between two points.

Theorems & Definitions (49)

  • Theorem 4.1: Symplecticity and Volume Preservation
  • proof
  • Theorem 5.1: Complexity Advantage of SGNs
  • proof
  • Proposition 5.2: Memory Complexity
  • proof
  • Theorem 5.3: Expressivity Comparison
  • proof
  • Proposition 5.4: Approximation Rate Comparison
  • proof
  • ...and 39 more