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Unlabeled sample compression schemes for oriented matroids

Tilen Marc

Abstract

A long-standing sample compression conjecture asks to linearly bound the size of the optimal sample compression schemes by the Vapnik-Chervonenkis (VC) dimension of an arbitrary class. In this paper, we explore the rich metric and combinatorial structure of oriented matroids (OMs) to construct proper unlabeled sample compression schemes for the classes of topes of OMs bounded by their VC-dimension. The result extends to the topes of affine OMs, as well as to the topes of the complexes of OMs that possess a corner peeling. The main tool that we use are the solutions of certain oriented matroid programs.

Unlabeled sample compression schemes for oriented matroids

Abstract

A long-standing sample compression conjecture asks to linearly bound the size of the optimal sample compression schemes by the Vapnik-Chervonenkis (VC) dimension of an arbitrary class. In this paper, we explore the rich metric and combinatorial structure of oriented matroids (OMs) to construct proper unlabeled sample compression schemes for the classes of topes of OMs bounded by their VC-dimension. The result extends to the topes of affine OMs, as well as to the topes of the complexes of OMs that possess a corner peeling. The main tool that we use are the solutions of certain oriented matroid programs.
Paper Structure (7 sections, 10 theorems, 6 equations, 5 figures, 1 table)

This paper contains 7 sections, 10 theorems, 6 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary definitions and first results
  3. OMs and partial cubes
  4. Affine OMs, oriented matroid programming, and corners
  5. Reconstructible maps
  6. Main result
  7. Acknowledgment

Key Result

Theorem 2.3

Let ${\mathcal{A}}=(\mathcal{M},g)$ be an affine OM, a halfspace of an OM $\mathcal{M}=(U,{\mathcal{L}}')$, $f\in U, f\neq g$, and $P(S)$ a non-empty polyhedron for some $S\in \{+,-\}^V, V\subseteq U$, bounded in the direction of $f$. Then the OM program $(\mathcal{M},g,f,P(S))$ has an optimal solut

Figures (5)

  • Figure 1: Concepts and samples coming from linear classifiers
  • Figure 2: Correspondence between sign vectors (covectors) of an OM $\mathcal{M}$ and its topological representation. The tope graph of $\mathcal{M}$ can be seen as a graph embedded in a hypercube with vertices $\{+,-\}^n$.
  • Figure 3: The orientation of lines in a rank 2 affine OM ${\mathcal{A}}$ with respect to a pseudo-hyperplane $f$ resulting in a graph on cocircuits of ${\mathcal{A}}$. Cocircuit $X$ is an optimal solution to the OM program with respect to polyhedron $P(S)$ (intersection of half-spaces) in direction of $f$. Furthermore, $D$ is a corner of the tope graph induced by ${\mathcal{T}}(X)$, such that ${\mathcal{T}}(X) \cap P(S) \subseteq D$, see Lemma \ref{['lem:orient_corner']}.
  • Figure 4: Defining a reconstructible map for the tope graph of an OM: cut a corner of the tope graph of an OM as in (a), define a reconstructible map for the remaining tope graph of an affine OM by Proposition \ref{['prop:afine']} as in (b), and extend the map to the OM by Proposition \ref{['prop:extend']}.
  • Figure 5: The tope graph of a COM with VC-dimension 2 having corners $D_1,\ldots,D_6$.

Theorems & Definitions (23)

  • Definition 1.1
  • Definition 2.1
  • Claim 2.2
  • proof
  • Theorem 2.3: Theorem 10.1.13 in bjvestwhzi-93
  • Claim 2.4
  • Lemma 2.5: Theorem 8 and Lemma 11 in Chapter 9(iii) of Man-82
  • Lemma 2.6
  • proof
  • Definition 2.7
  • ...and 13 more