论文标题
随机初始化的一层神经网络使数据可分离
Randomly Initialized One-Layer Neural Networks Make Data Linearly Separable
论文作者
论文摘要
最近，神经网络在将两个任意集映射到两个线性可分离的集合中表现出显着的功能。与受训练的网络相比，由于计算效率，通过随机初始化的神经网络实现这一目标的前景特别有吸引力。本文通过确定鉴于足够的宽度，一个随机初始初始化的单层神经网络可以以很高的概率可以将两个集合转换为两个线性可分离的集合而无需任何训练。此外，我们在神经网络的必要宽度上提供了这种现象的精确界限。我们的初始结合表现出对输入维度的指数依赖性，同时保持对所有其他参数的多项式依赖性。相比之下，我们的第二个界限独立于输入维度，有效地掩盖了维度的诅咒。我们证明中使用的主要工具在很大程度上依赖于几何原理和随机矩阵的浓度的融合。
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.
