基于受控向量场的参数高效生成建模
Parameter-Efficient Generative Modeling with Controlled Vector Fields
AI总结：
提出一种基于Chow-Rashevskii定理的连续时间生成建模框架，通过少量固定向量场和学习的标量控制构建表达流，实现参数高效的分布变换。
摘要：
受Chow-Rashevskii定理启发，我们引入了一个连续时间生成建模框架，该框架从一小组固定向量场和学习的标量控制中构建表达流。我们的框架不是学习无约束的高维向量场，而是通过学习标量控制函数来调制固定向量场，从而构造速度。当固定场是括号生成时，它们的李代数张成整个空间，提供了一种仅用少量学习控制通道即可实现表达性传输的机制，并为标准向量场参数化提供了一种参数高效的几何替代方案。这种解耦公式产生了一个结构化和可解释的生成模型，其中学习的标量输出通道的数量可以独立于环境维度选择。我们制定了一个表达性原则，表明在适当的可控性和适定性假设下，这种受控流可以将源分布传输到目标分布。我们使用连续归一化流似然目标训练所得模型，并在合成分布上进行了概念验证实验。
英文摘要：
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.
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