Summary of Weighted Sobolev Approximation Rates For Neural Networks on Unbounded Domains, by Ahmed Abdeljawad et al.
Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domains
by Ahmed Abdeljawad, Thomas Dittrich
First submitted to arxiv on: 6 Nov 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: Functional Analysis (math.FA); Machine Learning (stat.ML)
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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 This paper explores the capabilities of shallow neural networks in approximating functions in the spectral Barron space, specifically considering weighted Sobolev spaces. Building upon existing research, the authors focus on extending previous results to unbounded domains with decaying weights and Muckenhoupt weights over bounded domains. The study presents embedding results for weighted Fourier-Lebesgue spaces within weighted Sobolev spaces and establishes asymptotic approximation rates for shallow neural networks that avoid the curse of dimensionality. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary This paper looks at how well simple artificial neural networks can copy complex functions. It’s like trying to draw a picture using just a few basic shapes, but making sure you get it really close to what the original picture looks like. The researchers are trying to see if these simple networks can do this without getting worse as they try to copy more complicated pictures (that’s called the “curse of dimensionality”). They’re looking at two special cases: when the picture is all contained within a certain boundary, and when it can be as big or small as you want. The results show that these simple networks can get very close to copying the original functions without getting worse. |
Keywords
* Artificial intelligence * Embedding
