Symplectic Generative Networks (SGNs): A Hamiltonian Framework for Invertible Deep Generative Modeling
This work addresses the computational inefficiency in likelihood evaluation for generative models, offering a novel framework that could benefit researchers in machine learning and physics, though it is incremental as it builds on existing invertible models.
The paper tackles the problem of invertible deep generative modeling by introducing Symplectic Generative Networks (SGNs), which use Hamiltonian mechanics to create a volume-preserving mapping, enabling exact likelihood evaluation without Jacobian determinant calculations and providing theoretical guarantees like invertibility and universal approximation with error bounds.
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.