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

• URL: http://arxiv.org/abs/2505.22527v1
• Date: Wed, 28 May 2025 16:13:36 GMT
• Title: Symplectic Generative Networks (SGNs): A Hamiltonian Framework for Invertible Deep Generative Modeling
• Authors: Agnideep Aich, Ashit Aich, Bruce Wade,
• Abstract summary: We introduce the 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.<n>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 calculations.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce the 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.

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