Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Interior-point methods on manifolds: theory and applications

Hiroshi Hirai, Harold Nieuwboer, Michael Walter

TL;DR

This work gives a suitable generalization of self-concordance to Riemannian manifolds and shows that it gives the same structural results and guarantees as in the Euclidean setting, in particular local quadratic convergence of Newton's method.

Abstract

Interior-point methods offer a highly versatile framework for convex optimization that is effective in theory and practice. A key notion in their theory is that of a self-concordant barrier. We give a suitable generalization of self-concordance to Riemannian manifolds and show that it gives the same structural results and guarantees as in the Euclidean setting, in particular local quadratic convergence of Newton's method. We analyze a path-following method for optimizing compatible objectives over a convex domain for which one has a self-concordant barrier, and obtain the standard complexity guarantees as in the Euclidean setting. We provide general constructions of barriers, and show that on the space of positive-definite matrices and other symmetric spaces, the squared distance to a point is self-concordant. To demonstrate the versatility of our framework, we give algorithms with state-of-the-art complexity guarantees for the general class of scaling and non-commutative optimization problems, which have been of much recent interest, and we provide the first algorithms for efficiently finding high-precision solutions for computing minimal enclosing balls and geometric medians in nonpositive curvature.

Interior-point methods on manifolds: theory and applications

TL;DR

This work gives a suitable generalization of self-concordance to Riemannian manifolds and shows that it gives the same structural results and guarantees as in the Euclidean setting, in particular local quadratic convergence of Newton's method.

Abstract

Interior-point methods offer a highly versatile framework for convex optimization that is effective in theory and practice. A key notion in their theory is that of a self-concordant barrier. We give a suitable generalization of self-concordance to Riemannian manifolds and show that it gives the same structural results and guarantees as in the Euclidean setting, in particular local quadratic convergence of Newton's method. We analyze a path-following method for optimizing compatible objectives over a convex domain for which one has a self-concordant barrier, and obtain the standard complexity guarantees as in the Euclidean setting. We provide general constructions of barriers, and show that on the space of positive-definite matrices and other symmetric spaces, the squared distance to a point is self-concordant. To demonstrate the versatility of our framework, we give algorithms with state-of-the-art complexity guarantees for the general class of scaling and non-commutative optimization problems, which have been of much recent interest, and we provide the first algorithms for efficiently finding high-precision solutions for computing minimal enclosing balls and geometric medians in nonpositive curvature.
Paper Structure (32 sections, 77 theorems, 389 equations)

This paper contains 32 sections, 77 theorems, 389 equations.

Table of Contents

  1. Introduction and summary of results
  2. Self-concordance and Newton's method on manifolds
  3. Barriers and a path-following method on manifolds
  4. Examples of self-concordance: Squared distance in non-positive curvature
  5. Application I: Non-commutative optimization and scaling problems
  6. Application II: Minimum-enclosing ball problem on PD(n)
  7. Application III: Geometric median on hyperbolic space
  8. Organization of the paper
  9. Preliminaries in Riemannian geometry
  10. Metric, lengths, distances
  11. Covariant derivative and curvature
  12. Parallel transport, geodesics, completeness
  13. Gradient and Hessian
  14. Convexity
  15. Self-concordance and Newton's method on manifolds
  16. ...and 17 more sections

Key Result

Theorem 1.2

Let $f\colon D \to \mathbbm R$ be a strongly $\alpha$-self-concordant function defined on an open convex set $D\subseteq M$, with positive definite Hessian. Let $p\in D$ be a point such that $\lambda_{f,\alpha}(p) < 1$. Then the Newton iterate remains in the domain, i.e., $p_{f,+} \in D$, and moreov

Theorems & Definitions (147)

  • Definition 1.1: Self-concordance
  • Theorem 1.2: Quadratic convergence
  • Theorem 1.3: Path-following method
  • Theorem 1.4: Self-concordance of squared distance
  • Corollary 1.5
  • Theorem 1.6: Epigraph barrier
  • Theorem 1.7
  • Theorem 1.8: Non-commutative optimization
  • Theorem 1.9: Minimum enclosing ball
  • Theorem 1.10: Geometric median
  • ...and 137 more