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Using orthogonally structured positive bases for constructing positive $k$-spanning sets with cosine measure guarantees

Warren Hare, Gabriel Jarry-Bolduc, Sébastien Kerleau, Clément W. Royer

TL;DR

This paper investigates the construction of positive $k-spanning sets with geometrical guarantees, and provides a complete description, as well as a new notion of cosine measure that accounts for the resilient nature of such sets.

Abstract

Positive spanning sets span a given vector space by nonnegative linear combinations of their elements. These have attracted significant attention in recent years, owing to their extensive use in derivative-free optimization. In this setting, the quality of a positive spanning set is assessed through its cosine measure, a geometric quantity that expresses how well such a set covers the space of interest. In this paper, we investigate the construction of positive $k$-spanning sets with geometrical guarantees. Our results build on recently identified positive spanning sets, called orthogonally structured positive bases. We first describe how to identify such sets and compute their cosine measures efficiently. We then focus our study on positive $k$-spanning sets, for which we provide a complete description, as well as a new notion of cosine measure that accounts for the resilient nature of such sets. By combining our results, we are able to use orthogonally structured positive bases to create positive $k$-spanning sets with guarantees on the value of their cosine measures.

Using orthogonally structured positive bases for constructing positive $k$-spanning sets with cosine measure guarantees

TL;DR

This paper investigates the construction of positive $k-spanning sets with geometrical guarantees, and provides a complete description, as well as a new notion of cosine measure that accounts for the resilient nature of such sets.

Abstract

Positive spanning sets span a given vector space by nonnegative linear combinations of their elements. These have attracted significant attention in recent years, owing to their extensive use in derivative-free optimization. In this setting, the quality of a positive spanning set is assessed through its cosine measure, a geometric quantity that expresses how well such a set covers the space of interest. In this paper, we investigate the construction of positive -spanning sets with geometrical guarantees. Our results build on recently identified positive spanning sets, called orthogonally structured positive bases. We first describe how to identify such sets and compute their cosine measures efficiently. We then focus our study on positive -spanning sets, for which we provide a complete description, as well as a new notion of cosine measure that accounts for the resilient nature of such sets. By combining our results, we are able to use orthogonally structured positive bases to create positive -spanning sets with guarantees on the value of their cosine measures.
Paper Structure (16 sections, 18 theorems, 60 equations, 1 algorithm)

This paper contains 16 sections, 18 theorems, 60 equations, 1 algorithm.

Table of Contents

  1. Keywords
  2. 2020 Mathematics Subject Classification
  3. Introduction
  4. Notations
  5. Background on positive spanning sets
  6. Positive spanning sets and positive bases
  7. Orthogonally structured positive bases
  8. The cosine measure
  9. Computing OSPBs and their cosine measure
  10. Structure and detection of OSPBs
  11. Cosine measure of an orthogonally structured positive basis
  12. Positive TEXT-spanning sets and TEXT-cosine measure
  13. Positive TEXT-spanning property
  14. The TEXT-cosine measure
  15. Building positive TEXT-spanning sets using OSPBs
  16. ...and 1 more sections

Key Result

Lemma 2.1

Regis2016 Let $\mathcal{D}$ be a finite set of vectors in a subspace $\mathbb{L}$ of $\mathbb{R}^n$. The following statements are equivalent:

Theorems & Definitions (33)

  • Definition 2.1: Positive span and positive spanning set
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.2: Positive basis
  • Definition 2.3: Positive independence
  • Definition 2.4
  • Definition 2.5: Critical vectors
  • Theorem 2.1: Structure of a positive basis
  • Example 2.1
  • Definition 2.6: Orthogonally structured positive bases
  • ...and 23 more