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.

翻译：本文介绍了辛普莱克生成网络（SGN），这是一种深度生成模型，它利用哈密顿力学构建了潜在空间与数据空间之间的可逆、保体积映射。通过赋予潜在空间辛结构并将数据生成建模为哈密顿系统的时间演化，SGN实现了精确的似然评估，而无需承担雅可比行列式计算的计算开销。在本工作中，我们通过一个全面的理论框架为SGN提供了严谨的数学基础，该框架包括：（i）可逆性与保体积性的完整证明；（ii）与变分自编码器和归一化流的理论比较的形式化复杂度分析；（iii）带有定量误差界的强化通用逼近结果；（iv）基于统计流形几何的信息论分析；以及（v）带有自适应积分保证的广泛稳定性分析。这些贡献凸显了SGN的基本优势，并为未来对复杂高维数据的实证研究和应用奠定了坚实的基础。