2022/05/24 by Promit Ghosal, Ghosal, Promit, Srinath Mahankali +3 · 1 citation
Computer Science · #FOS: Computer and information sciences #FOS: Mathematics #Face and Expression Recognition #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Neural Networks and Applications #Probability (math.PR) #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2205.11716
openalex publication_date 2022/05/24 · openalex created_date 2022/05/27 · openalex updated_date 2026/07/28
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.