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-similar to standard assumptions in random matrix theory. 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 consider two types of noise, spherical and elliptical. In the spherical setting, we show that 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. The elliptical setting, which is important from a robustness standpoint, is less prone to easy interpretation.
In parallel work, REF considers a signalplus-noise model for X i and shows that the kernel matrix, in this case, approaches a kernel matrix which is based on the signal component alone.
On information plus noise kernel random matrices
Annals of Statistics 2010, Vol. 38, No. 5, 3191-3216
Mathematics