Asymptotic analysis of the Gaussian kernel matrix for partially noisy data in high dimensions

The Gaussian kernel is one of the most important kernels applicable to many research fields, including scientific computing and data science. In this paper, we present an asymptotic analysis of the Gaussian kernel matrix in high dimensions under a statistical model of noisy data. In 2010, Karoui showed an important asymptotic structure of the Gaussian kernel matrix under reasonable conditions on the random noise. In addition, consistency of the eigenvectors is proved, though the eigenvalues are inconsistent. This paper focuses on the above results and presents a consistent estimator of the Gaussian kernel matrix using the smallest eigenvalue, whenever the target matrix is low-rank in the asymptotic regime. This result is significant from an applied perspective because the Gaussian kernel matrix tends to be of low rank when some pairs of sample vectors are nearly identical, as can occur in large datasets. More importantly, this paper provides asymptotic analysis under a statistical model representing partial noise, where the structure of the asymptotic matrix makes it difficult to construct consistent estimators. To overcome this difficulty, we use a modified optimization method for constrained low-rank approximations, adapted to the asymptotic structure of the Gaussian kernel matrix. Our focus is on the perturbation structure of the submatrix used in the optimization, resulting in an estimator with strong consistency. To clarify the significance of our results, the paper systematically integrates key findings from existing research on Gaussian kernels for noisy data and low-rank matrix approximations, presenting them within a unified framework.