We study the spectral norm of random kernel matrices with polynomial scaling, where the number of samples scales polynomially with the data dimension. In this regime, Lu and Yau (2022) proved that the empirical spectral distribution converges to the additive free convolution of a semicircle law and a Marcenko-Pastur law. We demonstrate that the random kernel matrix can be decomposed into a "bulk" part and a low-rank part. The spectral norm of the "bulk" part almost surely converges to the edge of the limiting spectrum. In the special case where the random kernel matrices correspond to the inner products of random tensors, the empirical spectral distribution converges to the Marcenko-Pastur law. We prove that the largest and smallest eigenvalues converge to the corresponding spectral edges of the Marcenko-Pastur law.

翻译：我们研究了多项式缩放随机核矩阵的谱范数，其中样本数量随数据维度呈多项式增长。在此机制下，Lu和Yau（2022）证明了经验谱分布收敛于半圆律与Marcenko-Pastur律的加性自由卷积。我们证明随机核矩阵可分解为“主体”部分与低秩部分。“主体”部分的谱范数几乎必然收敛于极限谱的边缘。在随机核矩阵对应于随机张量内积的特殊情形下，经验谱分布收敛于Marcenko-Pastur律。我们证明了最大与最小特征值收敛于Marcenko-Pastur律的相应谱边缘。