We consider the problem of classifying data manifolds where each manifold represents invariances that are parameterized by continuous degrees of freedom. Conventional data augmentation methods rely on sampling large numbers of training examples from these manifolds. Instead, we propose an iterative algorithm, $MCP$, based on a cutting plane approach that efficiently solves a quadratic semi-infinite programming problem to find the maximum margin solution. We provide a proof of convergence as well as a polynomial bound on the number of iterations required for a desired tolerance in the objective function. The efficiency and performance of $MCP$ are demonstrated in high-dimensional simulations and on image manifolds generated from the ImageNet data set. Our results indicate that $MCP$ is able to rapidly learn good classifiers and shows superior generalization performance compared with conventional maximum margin methods using data augmentation methods.