Summary of Hardness Of Learning Neural Networks Under the Manifold Hypothesis, by Bobak T. Kiani et al.
Hardness of Learning Neural Networks under the Manifold Hypothesis
by Bobak T. Kiani, Jason Wang, Melanie Weber
First submitted to arxiv on: 3 Jun 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: Differential Geometry (math.DG); 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 investigates the impact of encoding geometric structure on the learnability of neural networks under the manifold hypothesis. The authors rigorously analyze the hardness of learning neural networks with minimal assumptions on the curvature and regularity of the manifold. They prove that learning is hard for manifolds of bounded curvature, but show that additional assumptions on the volume of the data manifold can guarantee learnability via interpolation. The paper also comments on intermediate regimes of manifolds which have heterogeneous features commonly found in real-world data. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary This paper looks at how well neural networks can learn when the data is thought to be lying near a low-dimensional surface, or “manifold”. Researchers already know that this kind of geometric structure can help improve learning, but they haven’t studied it rigorously. The authors try to figure out what assumptions we need to make about the manifold for learning to work well. They show that if the manifold is curved in certain ways, learning gets harder, but if there’s enough information in the data, learning becomes easier. This has implications for how we design and train neural networks. |
