Bounds on the smallest eigenvalue of the neural tangent kernel (NTK) are a key ingredient in the analysis of neural network optimization and memorization. However, existing results require distributional assumptions on the data and are limited to a high-dimensional setting, where the input dimension $d_0$ scales at least logarithmically in the number of samples $n$. In this work we remove both of these requirements and instead provide bounds in terms of a measure of the collinearity of the data: notably these bounds hold with high probability even when $d_0$ is held constant versus $n$. We prove our results through a novel application of the hemisphere transform.

翻译：神经正切核（NTK）最小特征值的界是分析神经网络优化与记忆机制的关键要素。然而，现有研究均需对数据分布施加假设，且仅限于高维设定——即输入维度$d_0$需随样本数$n$呈至少对数级增长。本研究同时突破了这两项限制，提出了一种基于数据共线性度量的界：特别值得注意的是，即使当$d_0$相对于$n$保持恒定，该界仍以高概率成立。我们通过半球变换的创新应用完成了相关证明。