arXiv:2412.05109v2 Announce Type: replace Abstract: We derive universal approximation results for the class of (countably) $m$-rectifiable measures. Specifically, we prove that $m$-rectifiable measures can be approximated as push-forwards of the one-dimensional Lebesgue measure on $[0,1]$ using ReLU neural networks with arbitrarily small approximation error in terms of Wasserstein distance. What is more, the weights in the networks under consideration are quantized and bounded and the number of ReLU neural networks required to achieve an approximation error of $\varepsilon$ is no larger than $
This research is part of the ongoing development in foundational AI, specifically in understanding the approximation capabilities of neural networks for complex mathematical structures.
It provides a theoretical underpinning for generative models and neural network capabilities, which is crucial for advancing AI's ability to create and manipulate high-dimensional data.
The improved theoretical understanding of neural networks' ability to generate rectifiable measures could lead to more efficient and robust generative AI architectures.
- · AI researchers
- · Generative AI companies
- · Machine learning framework developers
- · Traditional statistical modeling
Further theoretical advancements in neural network approximation capabilities.
Improved performance and broader application of generative AI models across various industries.
New forms of data synthesis and simulation becoming standard practice in fields reliant on complex data distributions.
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Read at arXiv cs.LG