On information plus noise kernel random matrices

Kernel random matrices have attracted a lot of interest in recent years, from both practical and theoretical standpoints. Most of the theoretical work so far has focused on the case were the data is sampled from a low-dimensional structure. Very recently, the first results concerning kernel random matrices with high-dimensional input data were obtained, in a setting where the data was sampled from a genuinely high-dimensional structure. In this paper, we consider the case where the data is of the type “information+noise”. In other words, each observation is the sum of two independent elements: one sampled from a “low-dimensional” structure, the signal part of the data, the other being high-dimensional noise, normalized to not overwhelm but still affect the signal. We show that in this setting the spectral properties of kernel random matrices can be understood from a new kernel matrix, computed only from the signal part of the data, but using (in general) a slightly different kernel. The Gaussian kernel has some special properties in this setting.