用哈密顿力学构建可逆生成模型，精确计算概率且无需繁琐雅可比计算。
Symplectic Generative Networks (SGNs): A Hamiltonian Framework for Invertible Deep Generative Modeling
- 基于哈密顿系统设计可逆映射，保持体积不变性。
- 理论证明模型可逆、稳定，计算复杂度优于变分自编码器和归一化流。
- 适合对生成模型数学基础与高维数据建模感兴趣的学者。
我们提出对称生成网络（SGN），一种利用哈密顿力学构建从隐空间到数据空间的可逆、保体积映射的深度生成模型。通过在隐空间引入辛结构，并将数据生成建模为哈密顿系统的时序演化，SGN 实现了无需计算雅可比行列式即可精确评估似然。本文提供了完整的理论框架，包括：(i) 可逆性与保体积性的严格证明；(ii) 与变分自编码器和归一化流的理论复杂度对比分析；(iii) 带定量误差界的新通用逼近定理；(iv) 基于统计流形几何的信息论分析；(v) 带自适应积分保证的稳定性分析。这些成果凸显了SGN的根本优势，为未来在复杂高维数据上的实证研究与应用奠定了坚实基础。
原文摘要 · Abstract (English)
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.
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