Kernel Polytope Faces Pursuit

Abstract. Polytope Faces Pursuit (PFP) is a greedy algorithm that ap-proximates the sparse solutions recovered by `1 regularised least-squares (Lasso) [4, 10] in a similar vein to (Orthogonal) Matching Pursuit (OMP) [16]. The algorithm is based on the geometry of the polar polytope where at each step a basis function is chosen by finding the maximal vertex us-ing a path-following method. The algorithmic complexity is of a similar order to OMP whilst being able to solve problems known to be hard for (O)MP. Matching Pursuit was extended to build kernel-based solu-tions to machine learning problems, resulting in the sparse regression algorithm, Kernel Matching Pursuit (KMP) [17]. We develop a new al-gorithm to build sparse kernel-based solutions using PFP, which we call Kernel Polytope Faces Pursuit (KPFP). We show the usefulness of this algorithm by providing a generalisation error bound [7] that takes into account a natural regression loss and experimental results on several benchmark datasets