arXiv:2511.22882v3 Announce Type: replace Abstract: We introduce boundary quotients and present a framework for learning densities on manifolds that arise as boundary quotients of simpler domains. We show that this framework can be used to construct normalizing flows on quotient manifolds $N/G$, where a discrete group $G$ acts on $N$. We instantiate this construction for genus-$g$ surfaces $\Sigma_g$. When $G$ is finite, we show applicability to symmetry aware learning; we demonstrate this on cyclic quotients of the 3-sphere. Experiments on lens spaces show that simple pre-quotient RealNVP mod
This paper represents continued academic progress in the mathematical and computational foundations of AI, specifically in normalizing flows for complex data structures like manifolds, which is a continuously evolving field.
Advanced techniques for learning densities on complex topological spaces could enable more robust and efficient AI models for tasks involving geometric data, impacting areas like robotics, computer vision, and scientific computing.
This introduces a formalized framework and new methodologies for constructive normalizing flows on quotient manifolds, potentially expanding the applicability and performance of generative models and density estimation.
- · AI researchers (mathematical AI, generative models)
- · Robotics
- · Computer vision
- · Scientific computing
Improved generative models for data with symmetries or manifold structures lead to more accurate simulations and analyses.
Better understanding and handling of complex data distributions could accelerate scientific discovery and engineering design.
The integration of these advanced mathematical concepts into commercial AI platforms could eventually enable more sophisticated autonomous systems and data analysis tools.
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