We welcome you to participate in the 5th NIPS Workshop on Optimization for Machine Learning, to be held at:
Lake Tahoe, Nevada, USA, Dec 8th, 2012
- Pablo Parrilo (MIT)
Semidefinite Optimization and Convex Algebraic Geometry
- Francis Bach (INRIA)
Sharp analysis of low-rank kernel matrix approximation
We consider supervised learning problems within the positive-deﬁnite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to inﬁnite-dimensional feature spaces, a common practical limiting difﬁculty is the necessity of computing the kernel matrix, which most frequently leads to algorithms with running time at least quadratic in the number of observations n, i.e., O(n^2). Low-rank approximations of the kernel matrix are often considered as they allow the reduction of running time complexities to O(p^2 n), where p is the rank of the approximation. The practicality of such methods thus depends on the required rank p. In this talk, I will show that for approximations based on a random subset of columns of the original kernel matrix, the rank p may be chosen to be linear in the degrees of freedom associated with the problem, a quantity which is classically used in the statistical analysis of such methods, and is often seen as the implicit number of parameters of non-parametric estimators. This result enables simple algorithms that have subquadratic running time complexity, but provably exhibit the same predictive performance than existing algorithms.