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Kernel methods are an extremely popular set of techniques used for manyimportant machine learning and data analysis applications. In addition tohaving good practical performances, these methods are supported by awell-developed theory. Kernel methods use an implicit mapping of the input datainto a high dimensional feature space defined by a kernel function, i.e., afunction returning the inner product between the images of two data points inthe feature space. Central to any kernel method is the kernel matrix, which isbuilt by evaluating the kernel function on a given sample dataset.  In this paper, we initiate the study of non-asymptotic spectral theory ofrandom kernel matrices. These are n x n random matrices whose (i,j)th entry isobtained by evaluating the kernel function on $x_i$ and $x_j$, where$x_1,...,x_n$ are a set of n independent random high-dimensional vectors. Ourmain contribution is to obtain tight upper bounds on the spectral norm (largesteigenvalue) of random kernel matrices constructed by commonly used kernelfunctions based on polynomials and Gaussian radial basis.  As an application of these results, we provide lower bounds on the distortionneeded for releasing the coefficients of kernel ridge regression underattribute privacy, a general privacy notion which captures a large class ofprivacy definitions. Kernel ridge regression is standard method for performingnon-parametric regression that regularly outperforms traditional regressionapproaches in various domains. Our privacy distortion lower bounds are thefirst for any kernel technique, and our analysis assumes realistic scenariosfor the input, unlike all previous lower bounds for other release problemswhich only hold under very restrictive input settings.

Spectral Norm of Random Kernel Matrices with Applications to Privacy