On a manifold formulation of self-concordant functions

TL;DR

This paper extends the theory of self-concordant functions to Riemannian manifolds, demonstrating properties like quadratic convergence and providing examples such as hyperbolic space distance functions.

Contribution

It formulates self-concordance on manifolds via covariant derivatives and shows that key properties extend, including examples in hyperbolic space.

Findings

Squared distance in hyperbolic space is self-concordant.

Logarithmic barrier of a ball in hyperbolic space is self-concordant.

Application to minimum enclosing ball problem in hyperbolic space.

Abstract

In this paper, we address an extension of the theory of self-concordant functions for a manifold. We formulate the self-concordance of a geodesically convex function by a condition of the covariant derivative of its Hessian, and verify that many of the analogous properties, such as the quadratic convergence of Newton's method and the polynomial iteration complexity of the path-following method, are naturally extended. However it is not known whether a useful class of self-concordant functions/barriers really exists for non-Euclidean manifolds. To this question, we provide a preliminary result that the squared distance function in the hyperbolic space of curvature $- \kappa$ is $\sqrt{\kappa}/2$-self-concordant and the associated logarithmic barrier of a ball of radius $R$ is an $O(\kappa R^2)$-self-concordant barrier. We also give an application to the minimum enclosing ball in a hyperbolic space.