高维分类器可用三层ReLU网络高效逼近，突破维度瓶颈。
High-dimensional classification problems with Barron regular boundaries under margin conditions
- 用三隐层ReLU网络逼近巴龙正则边界
- 强间隔条件下逼近率接近低维光滑函数水平
- 适合高维非光滑分类问题研究者参考
在假设间隔条件成立的前提下，我们证明具有巴龙正则决策边界的分类器可通过三隐藏层的ReLU神经网络以高阶多项式速率被逼近。特别地，在强间隔条件下，高维不连续分类器的逼近速率通常仅在近似低维光滑函数时才能达到。我们展示了这些表达率界如何导出接近 $n^{-1}$ 的快速学习界，其中 $n$ 为样本数量。此外，我们在具有不同间隔的二分类问题上进行了全面数值实验，涵盖三种不同维度，最高维度对应于MNIST数据集中的图像。
原文摘要 · Abstract (English)
We prove that a classifier with a Barron-regular decision boundary can be approximated with a rate of high polynomial degree by ReLU neural networks with three hidden layers when a margin condition is assumed. In particular, for strong margin conditions, high-dimensional discontinuous classifiers can be approximated with a rate that is typically only achievable when approximating a low-dimensional smooth function. We demonstrate how these expression rate bounds imply fast-rate learning bounds that are close to $n^{-1}$ where $n$ is the number of samples. In addition, we carry out comprehensive numerical experimentation on binary classification problems with various margins. We study three different dimensions, with the highest dimensional problem corresponding to images from the MNIST data set.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。