Summary of Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operators, by Unique Subedi et al.
Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operators
by Unique Subedi, Ambuj Tewari
First submitted to arxiv on: 16 Aug 2024
Categories
- Main: Machine Learning (stat.ML)
- Secondary: Machine Learning (cs.LG); Numerical Analysis (math.NA)
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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 The paper explores the theoretical foundations of operator learning, focusing on the linear layer of the Fourier Neural Operator architecture. The authors identify three primary errors that occur during the learning process: statistical error due to finite sample size, truncation error from finite rank approximation of the operator, and discretization error from handling functional data on a finite grid. To address these issues, the researchers develop a Discrete Fourier Transform (DFT) based least squares estimator and provide both upper and lower bounds on the errors. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary The paper looks at how we can learn operators using neural networks. It finds three main problems that happen when we do this: we might not have enough data, our operator might be simplified in a way that doesn’t match reality, or we’re only looking at a small part of the problem. To fix these issues, the authors create a special kind of estimator that uses something called the Discrete Fourier Transform. |
