Summary of Generating Rectifiable Measures Through Neural Networks, by Erwin Riegler et al.
Generating Rectifiable Measures through Neural Networks
by Erwin Riegler, Alex Bühler, Yang Pan, Helmut Bölcskei
First submitted to arxiv on: 6 Dec 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: Information Theory (cs.IT); Probability (math.PR); Statistics Theory (math.ST); 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 The abstract discusses universal approximation results for a class of measures called (countably) m-rectifiable measures. Specifically, it proves that these measures can be approximated using ReLU neural networks with arbitrarily small error in terms of Wasserstein distance. The weights in the networks are quantized and bounded, and the number of networks required to achieve an approximation error of ε is no larger than 2^(b(ε)) with b(ε) = O(ε^(-m)*log^2(ε)). This result improves upon a previous lemma by showing that the rate at which b(ε) tends to infinity as ε tends to zero equals the rectifiability parameter m, which can be much smaller than the ambient dimension. The paper extends this result to countably m-rectifiable measures and shows that this rate still equals the rectifiability parameter m provided certain technical assumptions are met. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary This research paper proves that a special type of measure can be approximated using neural networks with very small errors. Neural networks are computer systems that can learn from data, and they are used to make predictions or decisions. The measures in this paper are called m-rectifiable, which is a technical term. Essentially, the researchers show that these measures can be approximated by pushing forward one-dimensional Lebesgue measure onto [0,1] using ReLU neural networks. This has important implications for many areas of mathematics and computer science. |
Keywords
» Artificial intelligence » Relu
