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A Newton-Kantorovich Inverse Function Theorem in Quasi-Metric Spaces

Titus Pinta

TL;DR

The paper develops a framework for solving root-finding problems on quasi-metric spaces, motivated by applications in biology and phylogenetics. It introduces Newton differentiability as a nonsmooth calculus tool and builds Newton-type fixed-point methods using inversely compatible pseudo-linear maps, proving a Kantorovich-type inverse function theorem in this setting. The authors establish calculus rules for Newton differentiability, derive convergence rates including superlinear and higher, and illustrate the theory with a numerical example on a tree-like metric space. The work extends optimization on non-symmetric spaces, offering a pathway to handle nonsmooth problems in Euclidean subsets and guiding future algebraic development for metric-space optimization.

Abstract

The purpose of this work is to investigate root finding problems defined on (quasi-)metric spaces, and ranging in Euclidean spaces. The motivation for this line of inquiry stems from recent models in biology and phylogenetics, where problems of great practical significance are cast as optimization problems on (quasi-)metric spaces. We investigate a minimal algebraic setup that allows us to study a notion of differentiability suitable for Newton-type methods, called Newton differentiability. This notion of differentiability benefits from calculus rules and is sufficient to prove superlinear convergence of a Newton-type method. Finally, a Newton-Kantorovich-type theorem provides an inverse function result, applicable on (quasi-)metric spaces.

A Newton-Kantorovich Inverse Function Theorem in Quasi-Metric Spaces

TL;DR

The paper develops a framework for solving root-finding problems on quasi-metric spaces, motivated by applications in biology and phylogenetics. It introduces Newton differentiability as a nonsmooth calculus tool and builds Newton-type fixed-point methods using inversely compatible pseudo-linear maps, proving a Kantorovich-type inverse function theorem in this setting. The authors establish calculus rules for Newton differentiability, derive convergence rates including superlinear and higher, and illustrate the theory with a numerical example on a tree-like metric space. The work extends optimization on non-symmetric spaces, offering a pathway to handle nonsmooth problems in Euclidean subsets and guiding future algebraic development for metric-space optimization.

Abstract

The purpose of this work is to investigate root finding problems defined on (quasi-)metric spaces, and ranging in Euclidean spaces. The motivation for this line of inquiry stems from recent models in biology and phylogenetics, where problems of great practical significance are cast as optimization problems on (quasi-)metric spaces. We investigate a minimal algebraic setup that allows us to study a notion of differentiability suitable for Newton-type methods, called Newton differentiability. This notion of differentiability benefits from calculus rules and is sufficient to prove superlinear convergence of a Newton-type method. Finally, a Newton-Kantorovich-type theorem provides an inverse function result, applicable on (quasi-)metric spaces.

Paper Structure

This paper contains 11 sections, 14 theorems, 110 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and Basic Properties
  3. Algebraic Constructs
  4. Newton Differentiability
  5. Calculus of Newton Differentiability
  6. Newton-type Methods
  7. Higher Convergence Rates
  8. The Kantorovich-type Theorem
  9. A Numerical Example
  10. Conclusions and Limitations
  11. Acknowledgments

Key Result

Lemma 1.6

Let $\bar{x} \in \mathbb{M}$ and ${\{r_k\}}_{k \in \mathbb{N}}$ be a sequence of real numbers. If $\lim_{k \to \infty}r_k = 0$, then

Figures (1)

  • Figure 1: An example of the metric space $\mathbb{M}$ constructed using a binary tree. The points ${\color{red}(r, 0)}$, ${\color{blue}(b_x, .6)}$ and ${\color{orange}(b_y, .75)}$ are marked, together with the shortest path $b_x, b_1, b_2, b_y$ between $b_x$ and $b_y$

Theorems & Definitions (64)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3: Distance between Points and Subsets
  • Definition 1.4
  • Definition 1.5
  • Lemma 1.6
  • proof
  • Definition 1.7
  • Definition 1.8: Convergence Rates
  • Definition 1.9
  • ...and 54 more