Stable Autonomous Flow Matching
This work addresses a gap in the literature for machine learning applications with stable data points, though it appears incremental in linking existing control theory tools to generative models.
The paper tackles the problem of connecting control theory with flow matching generative models for physically stable data, characterizing which models are amenable to stochastic stability analysis and demonstrating theoretical results on examples.
In contexts where data samples represent a physically stable state, it is often assumed that the data points represent the local minima of an energy landscape. In control theory, it is well-known that energy can serve as an effective Lyapunov function. Despite this, connections between control theory and generative models in the literature are sparse, even though there are several machine learning applications with physically stable data points. In this paper, we focus on such data and a recent class of deep generative models called flow matching. We apply tools of stochastic stability for time-independent systems to flow matching models. In doing so, we characterize the space of flow matching models that are amenable to this treatment, as well as draw connections to other control theory principles. We demonstrate our theoretical results on two examples.