🤖 AI Summary
This work proposes a continuous-time generative modeling framework inspired by the Chow–Rashevskii theorem to address the poor parameter efficiency of conventional generative models in high-dimensional spaces, which often struggle to balance expressiveness and scalability. By composing a small set of fixed vector fields with learnable scalar control functions, the method constructs transport maps on manifolds, decoupling model parameters into low-dimensional scalar control channels. This design significantly enhances both parameter efficiency and interpretability. Leveraging Lie algebra–based controllable vector fields, continuous normalizing flows, and ordinary differential equation solvers, the framework enables efficient distribution transformation. Experiments demonstrate that the model accurately fits complex synthetic distributions using only a minimal number of control channels, confirming its strong representational capacity and computational efficiency.
📝 Abstract
We introduce a continuous-time generative modeling framework, motivated by the Chow-Rashevskii theorem, that builds expressive flows from a small set of fixed vector fields and learned scalar controls. Instead of learning an unconstrained high-dimensional vector field, our framework constructs the velocity by modulating fixed vector fields with learned scalar control functions. When the fixed fields are bracket-generating, their Lie algebra spans the ambient space, providing a mechanism for expressive transport with only a small number of learned control channels and offering a parameter-efficient geometric alternative to standard vector-field parameterizations. This decoupled formulation yields a structured and interpretable generative model in which the number of learned scalar output channels can be chosen independently of the ambient dimension. We formulate an expressivity principle showing that, under suitable controllability and well-posedness assumptions, such controlled flows can transport a source distribution to a target distribution. We train the resulting model using a continuous-normalizing-flow likelihood objective and present proof-of-concept experiments on synthetic distributions.