Summary of Basis-to-basis Operator Learning Using Function Encoders, by Tyler Ingebrand et al.
Basis-to-Basis Operator Learning Using Function Encoders
by Tyler Ingebrand, Adam J. Thorpe, Somdatta Goswami, Krishna Kumar, Ufuk Topcu
First submitted to arxiv on: 30 Sep 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: None
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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 introduces Basis-to-Basis (B2B) operator learning, a novel approach for learning operators on Hilbert spaces of functions. It decomposes the task into learning basis functions and a potentially nonlinear mapping between coefficients. B2B circumvents challenges in prior works by leveraging least squares to compute coefficients. The method is particularly effective for linear operators, where it computes a single matrix transformation with a closed-form solution. Additionally, the paper derives operator learning algorithms analogous to eigen-decomposition and singular value decomposition using deep theoretical connections between function encoders and functional analysis. Empirical validation on seven benchmark tasks shows B2B operator learning achieves a two-orders-of-magnitude improvement in accuracy over existing approaches. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary This paper creates a new way to learn operators for functions. It breaks down the task into two parts: learning special sets of basis functions for input and output spaces, and finding a connection between these basis functions. The method is better than previous attempts because it uses classic techniques like least squares to find the right coefficients. For linear operators, this method finds a simple solution that can be calculated quickly. The paper also shows how to use this new approach with existing methods to improve accuracy. |
