arXiv:2604.07213v2 Announce Type: replace Abstract: High-dimensional data are often assumed to lie on lower-dimensional manifolds. We study how to construct diffusion processes on this data manifold using only point cloud samples and without access to charts, projections, or other geometric primitives. Here, we introduce Implicit Manifold-valued Diffusions (IMDs), a data-driven mathematical formalism for defining stochastic differential equations in the original high-dimensional space that describe drifting Brownian particles evolving intrinsically on the underlying manifold. Our construction
The continuous advances in AI and machine learning necessitate more sophisticated mathematical frameworks to handle high-dimensional data, pushing research into novel diffusion processes.
This research provides a fundamental mathematical formalism for improving how AI systems learn from and interpret complex, unstructured data, potentially leading to more accurate and generalizable AI models.
The ability to define diffusion processes on implicit manifolds directly from point cloud samples, without requiring explicit geometric primitives, opens new avenues for generative AI and data analysis.
- · AI researchers
- · Generative AI developers
- · Data scientists
- · Machine learning platform providers
- · Developers reliant on traditional, rigid data modeling techniques
Improved generative models capable of creating highly realistic and diverse data outputs.
Enhanced performance in fields requiring complex data understanding, such as drug discovery, materials science, and robotics.
Acceleration of research into more adaptable and less data-hungry AI systems that can learn from sparse or noisy information.
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Read at arXiv cs.LG