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On extreme points and representer theorems for the Lipschitz unit ball on finite metric spaces

Kristian Bredies, Jonathan Chirinos Rodriguez, Emanuele Naldi

TL;DR

This paper studies extreme points of the Lipschitz unit ball on a finite metric space $$(\mathcal{X},d)$$ valued in a strictly convex Banach space $$(\mathcal{Y},\|\cdot\|)$$, focusing on $Lip_0^1$. It proves that the extreme points are exactly the set $\mathcal{E}$ defined by saturated Lipschitz edges along chains from $x_0$ to each $x_i$. A representer theorem is established: any $y\in Lip_0^1$ can be expressed as $y=\sum_{i=1}^k \lambda_i y^i$ with $k\le n+1$ and $y^i\in ext(Lip_0^1)$, $\lambda_i\ge0$, $\sum_i\lambda_i=1$, independent of the ambient dimension. Consequently, the classical Minkowski–Carathéodory bound, which scales with the target dimension, is improved from $nd+1$ to $n+1$, enabling dimension-free representations and potential algorithmic gains in vector-valued Lipschitz regularization contexts.

Abstract

In this note, we provide a characterization for the set of extreme points of the Lipschitz unit ball in a specific vectorial setting. While the analysis of the case of real-valued functions is covered extensively in the literature, no information about the vectorial case has been provided up to date. Here, we aim at partially filling this gap by considering functions mapping from a finite metric space to a strictly convex Banach space that satisfy the Lipschitz condition. As a consequence, we present a representer theorem for such functions. In this setting, the number of extreme points needed to express any point inside the ball is independent of the dimension, improving the classical result from Carathéodory.

On extreme points and representer theorems for the Lipschitz unit ball on finite metric spaces

TL;DR

This paper studies extreme points of the Lipschitz unit ball on a finite metric space valued in a strictly convex Banach space , focusing on . It proves that the extreme points are exactly the set defined by saturated Lipschitz edges along chains from to each . A representer theorem is established: any can be expressed as with and , , , independent of the ambient dimension. Consequently, the classical Minkowski–Carathéodory bound, which scales with the target dimension, is improved from to , enabling dimension-free representations and potential algorithmic gains in vector-valued Lipschitz regularization contexts.

Abstract

In this note, we provide a characterization for the set of extreme points of the Lipschitz unit ball in a specific vectorial setting. While the analysis of the case of real-valued functions is covered extensively in the literature, no information about the vectorial case has been provided up to date. Here, we aim at partially filling this gap by considering functions mapping from a finite metric space to a strictly convex Banach space that satisfy the Lipschitz condition. As a consequence, we present a representer theorem for such functions. In this setting, the number of extreme points needed to express any point inside the ball is independent of the dimension, improving the classical result from Carathéodory.
Paper Structure (3 sections, 1 theorem, 20 equations)

This paper contains 3 sections, 1 theorem, 20 equations.

Table of Contents

  1. Introduction
  2. Extreme points and representer theorems
  3. Acknowledgements

Key Result

lemma 1

Let $y\in\operatorname{Lip}^1_0$. For every $i=1,..., \, n$, there exists $0=i_0, \, i_1,.., \, i_k=i$, $k\geq 1$, such that $\|y_{i_{j+1}}-y_{i_j}\|=d(x_{i_{j+1}},x_{i_j})$, for every $j=0,..., \, k-1$ if and only if there does not exist a nonempty subset $S\subset\{1,..., \,n\}$ such that $\|y_i-y

Theorems & Definitions (4)

  • lemma 1
  • proof
  • proof
  • proof