# Learning Data Manifolds with a Cutting Plane Method

## TL;DR

This paper introduces M_{CP}, an efficient cutting-plane algorithm for classifying data manifolds, which outperforms traditional data augmentation methods in high-dimensional and image datasets.

## Contribution

The paper proposes a novel iterative cutting-plane algorithm for data manifold classification, providing convergence proof and demonstrating superior performance over existing methods.

## Key findings

- M_{CP} converges with a polynomial bound on iterations.
- M_{CP} achieves better generalization than data augmentation methods.
- The method performs well on high-dimensional and ImageNet-derived datasets.

## Abstract

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 upon sampling large numbers of training examples from these manifolds; instead, we propose an iterative algorithm called M_{CP} based upon 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 M_{CP} are demonstrated in high-dimensional simulations and on image manifolds generated from the ImageNet dataset. Our results indicate that M_{CP} is able to rapidly learn good classifiers and shows superior generalization performance compared with conventional maximum margin methods using data augmentation methods.