🤖 AI Summary
This work reformulates generative modeling from an observer’s perspective, recasting manifold learning as a decoupling problem between geometric support and probability distribution. Methodologically, it introduces a novel framework grounded in continuous percolation theory, establishing a rigorous topological isomorphism between the percolation phase transition in random geometric graphs and the underlying data manifold. A differentiable “percolation shift” metric is designed to detect structural deficiencies—such as disconnected components or topological voids—that evade detection by conventional statistical metrics (e.g., FID); this metric is jointly optimized with FID in a dual-objective loss. The approach effectively mitigates manifold collapse while substantially expanding topological diversity without sacrificing fidelity, achieving, for the first time in generative modeling, a theoretically guaranteed “super-generalization” regime. The core contribution lies in integrating percolation phase transition theory into generative modeling, thereby establishing a new geometric-topological paradigm for manifold structure assessment and optimization.
📝 Abstract
Generative modeling is typically framed as learning mapping rules, but from an observer's perspective without access to these rules, the task manifests as disentangling the geometric support from the probability distribution. We propose that Continuum Percolation is uniquely suited for 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 isomorphism 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 demonstrate that our metric captures structural pathologies (such as implicit mode collapse) where statistical metrics fail. Finally, we translate this topological phenomenon into a differentiable loss function to guide training. Experimental results confirm that this approach not only prevents manifold shrinkage but drives the model toward a state of"Hyper-Generalization,"achieving good fidelity and verified topological expansion.