arXiv:2605.25383v1 Announce Type: cross Abstract: We consider graph diffusion processes constructed from finite i.i.d. samples drawn from an unknown manifold embedded in ambient Euclidean space, where the graph affinity is defined by an ambient Gaussian kernel matrix. We show that the manifold heat semigroup $Q_t = e^{t\Delta}$ can be approximated directly by iterating the graph transition matrix $P$, under only low regularity assumptions on the test function $f$, including the case $f \in L^\infty$. We bound $\| P^n f - Q_t f \|$ in $\infty$-norm, with the operator application to $f$ properly
This paper represents continued academic progress in the theoretical understanding and practical application of manifold learning and diffusion processes, a foundational area for advanced AI research.
Improved theoretical guarantees for learning manifold structures from data can lead to more robust and powerful machine learning algorithms, particularly in areas like dimensionality reduction, data generation, and complex system modeling.
This research provides deeper mathematical understanding and approximation bounds for learning continuous manifold dynamics from discrete graph structures, enabling more reliable machine learning model development based on these principles.
- · AI researchers
- · Data scientists
- · Machine learning startups
- · Sectors with high-dimensional data
More accurate and efficient manifold learning algorithms become available for practical use.
This could enhance the performance of generative models, anomaly detection, and data visualization techniques.
Improved fundamental AI capabilities may accelerate progress in complex scientific discovery and autonomous systems.
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Read at arXiv cs.LG