Symplectic Generative Networks (SGNs): A Hamiltonian Framework for Invertible Deep Generative Modeling
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911238344146944 |
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| author | Aich, Agnideep Aich, Ashit |
| author_facet | Aich, Agnideep Aich, Ashit |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_22527 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symplectic Generative Networks (SGNs): A Hamiltonian Framework for Invertible Deep Generative Modeling Aich, Agnideep Aich, Ashit Machine Learning 68T07, 37J39, 65P10, 62B10, 53D22, 94A17 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. |
| title | Symplectic Generative Networks (SGNs): A Hamiltonian Framework for Invertible Deep Generative Modeling |
| topic | Machine Learning 68T07, 37J39, 65P10, 62B10, 53D22, 94A17 |
| url | https://arxiv.org/abs/2505.22527 |