Summary of Data Complexity Estimates For Operator Learning, by Nikola B. Kovachki and Samuel Lanthaler and Hrushikesh Mhaskar
Data Complexity Estimates for Operator Learning
by Nikola B. Kovachki, Samuel Lanthaler, Hrushikesh Mhaskar
First submitted to arxiv on: 25 May 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: 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 In this paper, researchers delve into the theoretical foundations governing efficient operator learning. Operator learning is a new paradigm for approximating nonlinear operators using data-driven methods. Despite its empirical success, the conditions for efficient operator learning remain incomplete. The study investigates how many input/output samples are needed to achieve a desired accuracy ε. From an n-widths perspective, the paper makes two key contributions. It derives lower bounds on n-widths for general classes of Lipschitz and Fréchet differentiable operators, revealing a “curse of data-complexity” that requires exponential sample size in the inverse of the desired accuracy. The study also shows that “parametric efficiency” implies “data efficiency,” using the Fourier neural operator (FNO) as a case study. Specifically, it demonstrates that efficient operator learning is attainable in both parametric and data complexity. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary This paper explores how to efficiently learn nonlinear operators using data-driven methods. Researchers want to know how many samples are needed to achieve a certain level of accuracy. They studied the problem from a special perspective called n-widths, which helps understand the difficulty of learning different types of operators. The study shows that some classes of operators require a lot of data to learn accurately, but for simpler operators, it’s possible to use fewer data samples. |
