# Supervised Learning via Euler's Elastica Models

Tong Lin, Hanlin Xue, Ling Wang, Bo Huang, Hongbin Zha.

Year: 2015, Volume: 16, Issue: 111, Pages: 3637−3686

#### Abstract

This paper investigates the Euler's elastica (EE) model for high-dimensional supervised learning problems in a function approximation framework. In 1744 Euler introduced the elastica energy for a 2D curve on modeling torsion-free thin elastic rods. Together with its degenerate form of total variation (TV), Euler's elastica has been successfully applied to low- dimensional data processing such as image denoising and image inpainting in the last two decades. Our motivation is to apply Euler's elastica to high-dimensional supervised learning problems. To this end, a supervised learning problem is modeled as an energy functional minimization under a new geometric regularization scheme, where the energy is composed of a squared loss and an elastica penalty. The elastica penalty aims at regularizing the approximated function by heavily penalizing large gradients and high curvature values on all level curves. We take a computational PDE approach to minimize the energy functional. By using variational principles, the energy minimization problem is transformed into an Euler-Lagrange PDE. However, this PDE is usually high-dimensional and can not be directly handled by common low-dimensional solvers. To circumvent this difficulty, we use radial basis functions (RBF) to approximate the target function, which reduces the optimization problem to finding the linear coefficients of these basis functions. Some theoretical properties of this new model, including the existence and uniqueness of solutions and universal consistency, are analyzed. Extensive experiments have demonstrated the effectiveness of the proposed model for binary classification, multi-class classification, and regression tasks.