  Kernel methods are an extremely popular set of techniques used for many
important machine learning and data analysis applications. In addition to
having good practical performances, these methods are supported by a
well-developed theory. Kernel methods use an implicit mapping of the input data
into a high dimensional feature space defined by a kernel function, i.e., a
function returning the inner product between the images of two data points in
the feature space. Central to any kernel method is the kernel matrix, which is
built by evaluating the kernel function on a given sample dataset.
  In this paper, we initiate the study of non-asymptotic spectral theory of
random kernel matrices. These are n x n random matrices whose (i,j)th entry is
obtained 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. Our
main contribution is to obtain tight upper bounds on the spectral norm (largest
eigenvalue) of random kernel matrices constructed by commonly used kernel
functions based on polynomials and Gaussian radial basis.
  As an application of these results, we provide lower bounds on the distortion
needed for releasing the coefficients of kernel ridge regression under
attribute privacy, a general privacy notion which captures a large class of
privacy definitions. Kernel ridge regression is standard method for performing
non-parametric regression that regularly outperforms traditional regression
approaches in various domains. Our privacy distortion lower bounds are the
first for any kernel technique, and our analysis assumes realistic scenarios
for the input, unlike all previous lower bounds for other release problems
which only hold under very restrictive input settings.
