Recently, neural networks have demonstrated remarkable capabilities in mapping two arbitrary sets to two linearly separable sets. The prospect of achieving this with randomly initialized neural networks is particularly appealing due to the computational efficiency compared to fully trained networks. This paper contributes by establishing that, given sufficient width, a randomly initialized one-layer neural network can, with high probability, transform two sets into two linearly separable sets without any training. Moreover, we furnish precise bounds on the necessary width of the neural network for this phenomenon to occur. Our initial bound exhibits exponential dependence on the input dimension while maintaining polynomial dependence on all other parameters. In contrast, our second bound is independent of input dimension, effectively surmounting the curse of dimensionality. The main tools used in our proof heavily relies on a fusion of geometric principles and concentration of random matrices.

翻译：近期，神经网络在将任意两个集合映射为线性可分集合方面展现出卓越能力。采用随机初始化神经网络实现这一目标尤为引人关注，因其相比完全训练的神经网络具有计算效率优势。本文证明，在宽度足够大的条件下，随机初始化的单层神经网络无需任何训练即可高概率地将两个集合转化为线性可分集合。此外，我们给出了实现这一现象所需神经网络宽度的精确界限。我们的第一个界限在输入维度上呈现指数依赖，而在其他参数上保持多项式依赖；相比之下，第二个界限与输入维度无关，有效克服了维度灾难。本文证明所采用的核心工具深度融合了几何原理与随机矩阵集中性质。