Support vector machine training using matrix completion techniques

We combine interior-point methods and results from matrix completion theory in an approximate method for the large dense quadratic programming problems that arise in support vector machine training. The basic idea is to replace the dense kernel matrix with the maximum determinant positive definite completion of a subset of the entries of the kernel matrix. The resulting approximate kernel matrix has a sparse inverse and this property can be exploited to dramatically improve the efficiency of interior-point methods. If the sparsity pattern is chordal, the sparse inverse and its Cholesky factors are easily computed by efficient recursive algorithms. Numerical experiments with training sets of size up to 60000 show that the approximate interiorpoint approach is competitive with the libsvm package in terms of error rate and speed. The completion technique can also be applied to other dense convex optimization problems arising in machine learning, for example, in Gaussian process classification.