eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-08-13
898
914
10.4230/LIPIcs.APPROX-RANDOM.2015.898
article
Spectral Norm of Random Kernel Matrices with Applications to Privacy
Kasiviswanathan, Shiva Prasad
Rudelson, Mark
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 performance, 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 properties 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 using common kernel functions such as 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol040-approx-random2015/LIPIcs.APPROX-RANDOM.2015.898/LIPIcs.APPROX-RANDOM.2015.898.pdf
Random Kernel Matrices
Spectral Norm
Subguassian Distribution
Data Privacy
Reconstruction Attacks