Manifold Percolation: from generative model to Reinforce learning
This addresses the problem of evaluating and improving generative models for researchers and practitioners by providing a topological perspective, though it appears incremental as it builds on existing percolation theory.
The paper tackles the problem of analyzing the geometric support of data in generative modeling by proposing continuum percolation as a method to project high-dimensional density estimation onto a geometric counting problem. It shows that the Percolation Shift metric captures structural pathologies like implicit mode collapse where standard metrics fail, and experimental results confirm this approach prevents manifold shrinkage and fosters synergistic improvement in generation and decision making.
Generative modeling is typically framed as learning mapping rules, but from an observer's perspective without access to these rules, the task becomes disentangling the geometric support from the probability distribution. We propose that continuum percolation is uniquely suited to this support analysis, as the sampling process effectively projects high-dimensional density estimation onto a geometric counting problem on the support. In this work, we establish a rigorous correspondence between the topological phase transitions of random geometric graphs and the underlying data manifold in high-dimensional space. By analyzing the relationship between our proposed Percolation Shift metric and FID, we show that this metric captures structural pathologies, such as implicit mode collapse, where standard statistical metrics fail. Finally, we translate this topological phenomenon into a differentiable loss function that guides training. Experimental results confirm that this approach not only prevents manifold shrinkage but also fosters a form of synergistic improvement, where topological stability becomes a prerequisite for sustained high fidelity in both static generation and sequential decision making.