Ample Completions of Oriented Matroids and Complexes of Uniform Oriented Matroids

This paper considers completions of tope graphs of COMs (complexes of oriented matroids) to ample partial cubes of the same VC-dimension.We show that these exist for OMs (oriented matroids) and CUOMs (complexes of uniform oriented matroids).This implies that tope graphs of OMs and CUOMs satisfy the sample compression conjecture -one of the central open questions of learning theory.We conjecture that the tope graph of every COM can be completed to an ample partial cube without increasing the VC-dimension.OMs, COMs, and AMPs (viewed as isometric subgraphs of hypercubes) give rise to set families for which the VC-dimension has a particular significance: the VC-dimension of an AMP is the largest dimension of a cube of its cube complex, the VC-dimension of an OM is its rank, and the VC-dimension of a COM is the largest VC-dimension of its faces.Littlestone and Warmuth [25] introduced the sample compression technique for deriving generalization bounds in machine learning.Floyd and Warmuth [19] asked whether any set family of VC-dimension d has a sample compression scheme of size O(d).This question, known as the sample compression conjecture, remains one of the oldest open problems in machine learning.It was shown in [28] that labeled compression schemes of size O(2 d ) exist.Moran and Warmuth [27] designed labeled compression schemes of size d for ample set systems.Chalopin et al. [9] designed (stronger) unlabeled compression schemes of size d for maximum families and characterized such schemes for ample set systems.In view of the above, it was noticed in [30] and [27] that the sample compression conjecture would be solved if any set family of VC-dimension d can be completed to an ample (or maximum) set system of VC-dimension O(d) or covered by O(2 d ) ample set systems of VC-dimension O(d).This opens a perspective that apart from its application to sample compression, is interesting in its own right: ample completions of structured set families.This is extending a given set system to an ample system by adding sets.A natural problem is ample completions of set families defined by partial cubes (i.e., isometric subgraphs of hypercubes).In [14], we prove that any partial cube of VC-dimension 2 admits an ample completion of VC-dimension 2.Moreover, we give a set family of VC-dimension 2 which has no ample completion of the same VC-dimension.In the present paper, we give an example of a partial cube of VC-dimension 3 which cannot be completed to an ample set system of VC-dimension 3. Hence, in higher dimension we cannot complete all partial cubes without increasing the VC-dimension.In the light of the above perspective, one may ask if there exists a constant c such that every partial cube of VC-dimension d admits an ample completion of VC-dimension ≤ cd?Even stronger, we wonder if partial cubes of VC-dimension d admit an ample completion of VC-dimension d + c.Note that no such additive constant c exists for general set families [30].In [14], we perform the ample completion of a partial cube of VC-dimension 2 in two steps.First, we show that they can be completed to tope graphs of COMs and next we complete the resulting graphs into ample ones, in both cases, without increasing the VC-dimension.In the present article, we are interested in the completion of this intermediate class of partial cubes.For COMs, we are inclined to believe that the following stronger result holds:Conjecture 1.The tope graph of any COM of VC-dimension d has an ample completion of VC-dimension d.COMs of rank 2 have the nice property that their faces are uniform OMs.This is not longer true in higher dimensions: COMs whose faces are uniform OMs constitute a proper subclass of COMs (that we call CUOMs).In this paper, we prove that Conjecture 1 holds for all tope graphs of OMs and CUOMs.This proves that set families arising from topes of OMs and CUOMs satisfy the sample compression conjecture.In Fig. 1, we present an example of the tope graph of a CUOM of VC-dimension 3, which we further use as a running example. Preliminaries2.1.VC-dimension.Let S be a family of subsets of an m-element set U .A subset X of U is shattered by S if for all Y ⊆ X there exists S ∈ S such that S ∩ X = Y .The Vapnik-Chervonenkis dimension [35] (the VC-dimension for short) VC-dim(S) of S is the cardinality of the largest subset of U shattered by S. Any set family S ⊆ 2 U can be viewed as a subset of vertices of the m-dimensional hypercube Q m = Q(U ).Denote by G(S) the subgraph of Q m