Summary of Blessing Of Dimensionality For Approximating Sobolev Classes on Manifolds, by Hong Ye Tan et al.
Blessing of Dimensionality for Approximating Sobolev Classes on Manifolds
by Hong Ye Tan, Subhadip Mukherjee, Junqi Tang, Carola-Bibiane Schönlieb
First submitted to arxiv on: 13 Aug 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: Statistics Theory (math.ST)
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| Summary difficulty | Written by | Summary |
|---|---|---|
| High | Paper authors | High Difficulty Summary Read the original abstract here |
| Medium | GrooveSquid.com (original content) | Medium Difficulty Summary A recently published study empirically supports the manifold hypothesis, which posits that high-dimensional data is supported on or around a low-dimensional manifold. By assuming this hypothesis, researchers have derived bounds independent of any embedding space, leading to improved performance in very high dimensions. This paper contributes to existing theoretical results by providing statistical complexity bounds for approximating bounded Sobolev functions on compact manifolds, demonstrating that the required complexity is bounded from below and dependent only on the manifold’s intrinsic properties. These findings complement existing approximation results for ReLU networks on manifolds, which provide upper bounds on generalization capacity. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary This study explores the idea that high-dimensional data is actually supported on a low-dimensional surface called a manifold. Researchers have found that some methods work better than others when dealing with very high dimensions. To understand this better, they assume that the data lies on or near a manifold and try to find bounds that don’t rely on any specific way of projecting the data into a lower dimension. The study shows that the amount of information needed to approximate certain types of functions on a manifold is limited by the properties of the manifold itself. |
Keywords
» Artificial intelligence » Embedding space » Generalization » Relu
