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Ample completions of OMs and CUOMs

07/24/2020
by   Victor Chepoi, et al.
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This paper considers completions of COMs (complexes 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 OMs and CUOMs satisfy the sample compression conjecture – one of the central open questions of learning theory. We conjecture that every COM can be completed to an ample partial cube without increasing the VC-dimension.

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1. Introduction

Oriented matroids (OMs), co-invented by Bland and Las Vergnas [7] and Folkman and Lawrence [19]

, represent a unified combinatorial theory of orientations of ordinary matroids. They capture the basic properties of sign vectors representing the circuits in a directed graph and the regions in a central hyperplane arrangement in

. Oriented matroids are systems of sign vectors satisfying three simple axioms (composition, strong elimination, and symmetry) and may be defined in a multitude of ways, see the book by Björner et al. [6]. The tope graphs of OMs can be viewed as subgraphs of the hypercube satisfying two strong properties: they are centrally-symmetric and are isometric subgraphs of , i.e., are antipodal partial cubes [6].

Ample sets (AMPs) have been introduced by Lawrence [23] as asymmetric counterparts of oriented matroids and have been re-discovered independently by several works in different contexts [3, 8, 35]. Consequently, they received different names: lopsided [23], simple [35], extremal [8], and ample [3, 16]. Lawrence [23] defined ample sets for the investigation of the possible sign patterns realized by a convex set in . Ample sets admit a multitude of combinatorial and geometric characterizations [3, 8, 23] and comprise many natural examples arising from discrete geometry, combinatorics, and geometry of groups [3, 23]. Analogously to tope graphs of OMs, AMPs induce isometric subgraphs of . In fact, they satisfy a much stronger property: any two parallel cubes are connected in the set by a shortest path of parallel cubes.

Complexes of oriented matroids (COMs) have been introduced and investigated in [5] as a far-reaching natural common generalization of oriented matroids and ample sets. COMs are defined in a similar way as OMs, simply replacing the global axiom of symmetry by a local axiom of face symmetry. This simple alteration leads to a rich combinatorial and geometric structure that is build from OM faces but is quite different from OMs. Replacing each face by a PL-ball, each COM leads to a contractible cell complex (topologically, OMs are spheres and AMPs are contractible cubical complexes). The tope graphs of COMs are still isometric subgraphs of hypercubes; as such, they have been characterized in [21].

Set families are fundamental objects in combinatorics, algorithmics, machine learning, discrete geometry, and combinatorial optimization. The Vapnik-Chervonenkis dimension (

VC-dimension for short) of a set family was introduced by Vapnik and Chervonenkis [34] and plays a central role in the theory of PAC-learning. The VC-dimension was adopted in the above areas as a complexity measure and as a combinatorial dimension of the set family. The topes of OMs, COMs, and AMPs (viewed as isometric subgraphs of hypercubes) give raise 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 [24] introduced the sample compression technique for deriving generalization bounds in machine learning. Floyd and Warmuth [18] asked whether any set family of VC-dimension has a sample compression scheme of size . This question remains one of the oldest open problems in machine learning. It was shown in [27] that labeled compression schemes of size exist. Moran and Warmuth [26] designed labeled compression schemes of size for ample sets. Chalopin et al. [9] designed (stronger) unlabeled compression schemes of size for maximum families and characterized such schemes for ample sets. In view of all this, it was noticed in [29] and [26] that the sample compression conjecture would be solved if one can show that any set family of VC-dimension can be completed to an ample (or maximum) set of VC-dimension or can be covered by ample sets of VC-dimension .

This opens a perspective that apart from its application to sample compression, is interesting in its own right: ample completions of structured set families. A natural class are ample completions of set families defined by partial cubes (i.e., isometric subgraphs of hypercubes). In [13], we prove that any partial cube of VC-dimension 2 admits an ample completion of VC-dimension 2. On the other hand, 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 of VC-dimension 3. In the light of the above question, one may ask if there exists a constant such that every partial cube of VC-dimension admits an ample completion of VC-dimension ? Even stronger, we wonder if partial cubes of VC-dimension admit an ample completion of VC-dimension . Note that no such additive constant exists for general set families [29]. Finally, in case of COMs we are inclined to believe that the following stronger result holds:

Conjecture 1.

Any COM of VC-dimension has an ample completion of VC-dimension .

Here we prove that Conjecture 1 holds for all OMs and for all COMs whose faces are uniform OMs (we call them CUOMs). This proves that set families arising from topes of OMs and CUOMs satisfy the sample compression conjecture.

2. Preliminaries

2.1. VC-dimension

Let be a family of subsets of an -element set . A subset of is shattered by if for all there exists such that . The Vapnik-Chervonenkis dimension [34] (the VC-dimension for short) of is the cardinality of the largest subset of shattered by . Any set family can be viewed as a subset of vertices of the -dimensional hypercube . Denote by the subgraph of induced by the vertices of corresponding to the sets of ; is called the 1-inclusion graph of ; each subgraph of is the 1-inclusion graph of a family of subsets of . A subgraph of has VC-dimension if is the 1-inclusion graph of a set family of VC-dimension . For a subgraph of we denote by the smallest cube of containing .

A -cube of is the 1-inclusion graph of the set family , where is a subset of . If , then any -cube is a -dimensional subcube of and contains -cubes. We call any two -cubes parallel cubes. A subset of is strongly shattered by if the 1-inclusion graph of contains a -cube. Denote by and the families consisting of all shattered and of all strongly shattered sets of , respectively. Clearly, and both and are closed under taking subsets, i.e., and are abstract simplicial complexes. The VC-dimension of is thus the size of a largest set shattered by , i.e., the dimension of the simplicial complex .

Two important inequalities relate a set family with its VC-dimension. The first one, the Sauer-Shelah lemma [30, 31] establishes that if , then the number of sets in a set family with VC-dimension is upper bounded by . The second stronger inequality, called the sandwich lemma [2, 8, 16, 28], proves that is sandwiched between the number of strongly shattered sets and the number of shattered sets [2, 8, 16, 28], i.e., . If and , then cannot contain more than simplices, thus the sandwich lemma yields the Sauer-Shelah lemma. The set families for which the Sauer-Shelah bounds are tight are called maximum sets [20, 18] and the set families for which the upper bounds in the sandwich lemma are tight are called ample, lopsided, and extremal sets [3, 8, 23]. Every maximum set is ample, but not vice versa.

2.2. Partial cubes

All graphs in this paper are finite, connected, and simple. The distance between two vertices and is the length of a shortest -path, and the interval between and consists of all vertices on shortest -paths: An induced subgraph of is isometric if the distance between any pair of vertices in is the same as that in An induced subgraph of (or its vertex set ) is called convex if it includes the interval of between any two vertices of . A subset or the subgraph of induced by is called gated (in [17] if for every vertex outside there exists a vertex (the gate of ) in such that each vertex of is connected with by a shortest path passing through the gate . It is easy to see that if has a gate in , then it is unique and that gated sets are convex.

A graph is isometrically embeddable into a graph if there exists a mapping such that for all vertices , i.e., is an isometric subgraph of . A graph is called a partial cube if it admits an isometric embedding into some hypercube . For an edge of , let . For an edge , the sets and are called complementary halfspaces of .

Theorem 1.

[15] A graph is a partial cube if and only if is bipartite and for any edge the sets and are convex.

Djoković [15] introduced the following binary relation on the edges of : for two edges and , we set if and only if and . Under the conditions of the theorem, if and only if and , i.e. is an equivalence relation. Let be the equivalence classes of and let be an arbitrary vertex taken as the basepoint of . For a -class , let be the pair of complementary halfspaces of defined by setting and for an arbitrary edge such that . The isometric embedding of into the -dimensional hypercube is obtained by setting for any vertex .

2.3. OMs, COMs, and AMPs

We recall the basic notions and results from the theory of oriented matroids (OMs), of complexes of oriented matroids (COMs), and of ample sets (AMPs). We follow the book [6] for OMs, the paper [5] for COMs, and the papers [3, 23] for AMPs.

2.3.1. OMs: oriented matroids

Oriented matroids (OMs) are abstractions of sign vectors of the regions in a central hyperplane arrangement of . Let be a set with elements and let be a system of sign vectors, i.e., maps to . The elements of are referred to as covectors and denoted by capital letters , etc. For , the subset is called the support of and its complement the zero set of . For a sign vector and a subset , let be the restriction of to . Let be the product ordering on relative to the standard ordering of signs with and . For , we call the separator of and . The composition of and is the sign vector , where for all , if and if .

Definition 1.

An oriented matroid (OM) is a system of sign vectors satisfying

  • (Composition) for all .

  • (Strong elimination) for each pair and for each , there exists such that and for all .

  • (Symmetry) that is, is closed under sign reversal.

A system of sign-vectors is simple if it has no “redundant” elements, i.e., for each , and for each pair there exist with . We will only consider simple OMs, without explicitly stating it every time. The poset of an OM together with an artificial global maximum forms a graded lattice, called the big face lattice . The length of the maximal chains of minus one is called the rank of and denoted . Note that equals the rank of the underlying unoriented matroid [6, Theorem 4.1.14].

From (C), (Sym), and (SE) it follows that the set of topes of any simple OM are -vectors. Thus, can be viewed as a family of subsets of , where for each an element belongs to the corresponding set if and does not belong to the set otherwise. The tope graph of an OM is the 1-inclusion graph of the set of topes of . The Topological Representation Theorem of Oriented Matroids of [19] characterizes tope graphs of OMs as region graphs of pseudo-sphere arrangements in a sphere  [6], where is the rank of the OM. It is also well-known (see for example [6]) that tope graphs of OMs are partial cubes and that can be recovered from its tope graph (up to isomorphism). Therefore, we can define all terms in the language of tope graphs. Note also that topes of are the coatoms of .

Another important axiomatization of OMs is in terms of cocircuits. The cocircuits of are the minimal non-zero elements of , i.e., its atoms. The collection of cocircuits is denoted by and can be axiomatized as follows: a system of sign vectors is called an oriented matroid (OM) if satisfies (Sym) and the following two axioms:

  • (Incomparability) implies for all .

  • (Elimination) for each pair with and for each , there exists such that and for all .

The set of covectors can be derived from by taking the closure of under composition. The axiomatization of OMs via cocircuits is used to define uniform oriented matroids.

Definition 2.

[6] A uniform oriented matroid (UOM) of rank on a set of size is an OM such that consists of two opposite signings of each subset of of size .

2.3.2. COMs: complexes of oriented matroids

Complexes of oriented matroids (COMs) are abstractions of sign vectors of the regions of an arrangement of hyperplanes restricted to an open convex set of . COMs are defined in a similar way as OMs, simply replacing the global axiom (Sym) by a weaker local axiom (FS) of face symmetry:

Definition 3.

A complex of oriented matroids (COMs) is a system of sign vectors satisfying (SE) and the following axiom:

  • (Face symmetry) for all .

As for OMs, we restrict ourselves to simple COMs, i.e., COMs defining simple systems of sign vectors. One can see that (FS) implies (C), thus OMs are exactly the COMs containing the zero vector , see [5]. A COM is realizable if is the system of sign vectors of the regions in an arrangement of (oriented) hyperplanes restricted to an open convex set of . For other examples of (tope graphs of) COMs, see [5, 12, 21].

The simple twist between (Sym) and (FS) leads to a rich combinatorial and geometric structure that is build from OMs but is quite different from OMs. Let be a COM and be a covector of . The face of is ; a facet is a maximal proper face. From the definition, any face consists of the sign vectors of all faces of the subcube of with barycenter . By [5, Lemma 4], each face of is an OM. Since OMs are COMs, each face of an OM is an OM and the facets correspond to cocircuits. Furthermore, by [5, Section 11] replacing each combinatorial face of by a PL-ball, we obtain a contractible cell complex associated to each COM. The topes and the tope graphs of COMs are defined in the same way as for OMs. Again, the topes are -vectors, the tope graph is a partial cubes, and the COM can be recovered from its tope graph, see [5] or [21].

We continue with the definitions of AMPs and CUOMs. For , let .

Definition 4.

[5] An ample system (AMP) is a COM satisfying the following axiom:

  • (Ideal composition) .

Definition 5.

A complex of uniform oriented matroids (CUOM) is a COM in which each facet is an UOM.

2.4. Ample sets

Ample sets (AMPs) are abstractions of sign vectors of the regions in a central arrangement of axis-parallel hyperplanes restricted to an open convex set of . We already defined ample sets (1) as COMs such that and (2) as families of sets for which the upper bounds in the sandwich lemma are tight: . The set family arising in the second definition is the set of topes of the system of sign vectors from the first definition. As in case of OMs and COMs, we can consider AMPs as set families or as partial cubes.

By [3, 8], is ample if and only if and if and only if . This can be rephrased in the following combinatorial way: is ample if and only if each set shattered by is strongly shattered. Consequently, the VC-dimension of an ample family is the dimension of the largest cube in its 1-inclusion graph. A nice characterization of ample set families was provided in [23]: is ample if and only if for any cube of if is closed under taking antipodes, then either or is included in . The paper [3] provides metric and recursive characterizations of ample families. We continue with one of them.

Recall that any two -cubes of are called parallel cubes. The distance is the distance between mutually closest vertices of and . A gallery of length between and is a sequence of -cubes of such that is a cube for every . A geodesic gallery is a gallery of length .

Proposition 1.

[3] is ample if and only if any two parallel cubes of can be connected in by a geodesic gallery.

Thus, the 1-inclusion graph of an ample set is a partial cube and we will speak about ample partial cubes. We conclude with the definition of ample completions.

Definition 6.

An ample completion of a subgraph of VC-dimension of is an ample partial cube containing as a subgraph and such that .

3. Auxiliary results

In this section we recall or prove some auxiliary results used in the proofs.

3.1. Partial cubes and VC-dimension

In this subsection, we closely follow [12] and [13]. Let be a partial cube, isometrically embedded in the hypercube .

3.1.1. pc-Minors and VC-dimension

For a -class of a partial cube , an elementary restriction consists of taking one of the halfspaces and . More generally, a restriction is a convex subgraph of induced by the intersection of a set of halfspaces of . Since any convex subgraph of a partial cube is the intersection of halfspaces [1, 10], the restrictions of coincide with the convex subgraphs of . For a -class , the graph obtained from by contracting the edges of is called an (-)contraction of . For a vertex of , let be the image of under the -contraction. We will apply to subsets , by setting . In particular, we denote the -contraction of by . By [11, Theorem 3], is an isometric subgraph of , thus the class of partial cubes is closed under contractions. Since edge contractions in graphs commute, if are two distinct -classes, then . Consequently, for a set of -classes, we can denote by the isometric subgraph of obtained from by contracting the equivalence classes of edges from . Contractions and restrictions commute in partial cubes: any set of restrictions and contractions of a partial cube provide the same result, independently on the order they are performed in. The resulting partial cube is called a partial cube minor (or pc-minor) of . For a partial cube , let denote the class of all partial cubes not having as a pc-minor. For partial cubes, the VC-dimension can be formulated in terms of pc-minors:

Lemma 1.

[13, Lemma 1] A partial cube has VC-dimension if and only if .

An antipode of a vertex in a partial cube is a (necessarily unique) vertex such that . A partial cube is antipodal if all its vertices have antipodes. For a subgraph of an antipodal partial cube we denote by the set of antipodes of in . We will also use several times the following results (the last one is a direct consequence of Theorem 1):

Lemma 2.

[13, Lemma 7] If is a proper convex subgraph of an antipodal partial cube , then .

Lemma 3.

[21] Antipodal partial cubes are closed under contractions.

Lemma 4.

Intervals of partial cubes are convex.

3.1.2. Shattering via Cartesian products

The -cube is the Cartesian product of copies of , i.e., . For a subset of size , denote by the Cartesian product of the factors of indexed by the elements of ; clearly, is a -cube . Analogously, let be the -cube defined by . Then . For , let be the subgraph of induced by the sets ; each is isomorphic to the cube . Let be an isometric subgraph of . We also denote by the intersection of with the cube and we call the -fiber of . Since each is a convex subgraph of and is an isometric subgraph of , each fiber is either empty or a non-empty convex subgraph of . Then the definition of shattering can be rephrased in the following way:

Lemma 5.

A subset of is shattered by an isometric subgraph of if and only if each fiber , , is a nonempty convex subgraph of .

If is shattered by , we call the map such that for all , a shattering map. The edges of between two different fibers are called -edges. Note that for any -edge of there exists and such that corresponds to and corresponds to . Since the fibers define a partition of the vertex-set of , any path connecting two vertices from different fibers of contains -edges.

The following simple lemmas are well-known and will be used in our proofs:

Lemma 6.

A -path of a partial cube is a shortest path if and only if all edges of belong to different -classes of .

Proof.

Suppose and let and be edges of that are consecutive with respect to . Then belong to the same halfspace or and belong to the complementary halfspace. Since , this contradicts Theorem 1. ∎

Lemma 7.

If is a partial cube, a gated subgraph of , a vertex of , and the gate of in , then no shortest -path of contains edges of a -class of .

Proof.

Suppose a shortest -path contains an edge of a -class of . Let be an edge of belonging to . Since is bipartite, let . Since is the gate of in , the path constituted by , a shortest -path of , and the edge is a shortest -path of . Since contains two edges of , cannot be a shortest path. ∎

3.1.3. Isometric expansions and VC-dimension

A triplet is called an isometric cover of a connected graph , if the following conditions are satisfied:

  • and are two isometric subgraphs of ;

  • and ;

  • and is the subgraph of induced by .

A graph is an isometric expansion of with respect to an isometric cover of (notation ) if is obtained from by replacing each vertex of by a vertex and each vertex of by a vertex such that and , are adjacent in if and only if and are adjacent vertices of and is an edge of if and only if is a vertex of . If (and thus ), then the isometric expansion is called peripheral and we say that is obtained from by a peripheral expansion with respect to . Note that if is not peripheral, then is a separator of . By [10, 11], is a partial cube if and only if can be obtained by a sequence of isometric expansions from a single vertex.

There is an intimate relation between contractions and isometric expansions. If is a partial cube and is a -class of , then contracting we obtain the pc-minor of . Then can be obtained from by an isometric expansion with respect to , where and are the images by the contraction of the halfspaces and of and is the contraction of the vertices of incident to edges from .

Proposition 2.

[13, Proposition 5] Let be obtained from by an isometric expansion with respect to . Then if and only if .

3.2. OMs, COMs, and AMPs

Here we recall some results about OMs, COMs, and AMPs.

3.2.1. Faces

First, since OMs satisfy the axiom (Sym), we obtain:

Lemma 8.

Any OM is an antipodal partial cube.

Let be a COM. For a covector , recall that denotes the face of . Let also denote the smallest cube of containing . Note that and are defined by the same set of -classes.

Lemma 9.

[5] For each covector of a COM , the face is an OM.

The following lemma is implicit in [5] and explicit in [21]:

Lemma 10.

For each covector of a COM , the face is a gated subgraph of the tope graph of . Moreover, for any tope of , is the gate of in the cube .

Proof.

Pick any . By the definition of , , thus is a tope of . By definition of , belongs to (and thus to ). Since for all necessarily is the gate of in

3.2.2. Minors and pc-minors

We start with the following result about pc-minors of COMs and AMPs, which follows from the results of [3] and [5]:

Lemma 11.

The classes of COMs and AMPs are closed under taking pc-minors. The class of OMs is closed under contractions.

We continue with the notions of restriction, contraction, and minors for COMs (which can be compared with the similar notions for partial cubes). Let be a COM and . Given a sign vector by we refer to the restriction of to , that is with for all . The deletion of is defined as , where . The contraction of is defined as , where . If arises by deletions and contractions from , is said to be minor of . Deletion in a COM translates to pc-contraction in its tope graph, while contraction corresponds to what is called taking the zone graph, see [21].

Lemma 12.

[5, Lemma 1] The classes of COM and AMP are closed under taking minors.

3.2.3. Hyperplanes, carriers, and half-carriers

For a COM , a hyperplane of is the set The carrier of the hyperplane is the union of all faces of with . The positive and negative (open) halfspaces supported by the hyperplane are and The carrier minus splits into its positive and negative parts: and which we call half-carriers.

Proposition 3.

[5, Proposition 6] In COMs and AMPs, all halfspaces, hyperplanes, carriers, and half-carriers are COMs and AMPs. In OMs, all hyperplanes and carriers are OMs.

The result about AMPs was not stated in [5, Proposition 6], however it easily follows from the definition of AMPs as COMs satisfying the axiom (IC). The result also follows from [3].

3.2.4. Amalgams

One important property of COMs is that they all can be obtained by amalgams from their maximal faces, i.e., they are amalgams of OMs. Now we make this definition precise. Following [5], we say that a system of sign vectors is a COM-amalgam of two COMs and if the following conditions are satisfied:

  • with ;

  • is a COM;

  • and ;

  • for and with there exists a shortest path in the tope graph of the deletion of .

Proposition 4.

[5, Proposition 7 & Corollary 2] The COM-amalgam of two COMs is a COM in which every facet is a facet of or . Any COM is obtained via successive COM-amalgams from its maximal faces.

We will not make use of the following and state it here without proof just for completeness:

Corollary 1.

The COM-amalgam of two AMPs such that is ample is an AMP. Any AMP is obtained via successive AMP-amalgams from its maximal faces.

Now, we present a notion of AMP-amalgam formulated in terms of graphs. We say that a graph is an AMP-amalgam of and if is an isometric cover of and and are ample partial cubes. The main difference between this and COM-amalgams is that condition (4) in the definition of a COM-amalgam is replaced by the weaker condition that is a partial cube. The next result was proved in [4] but never published:

Proposition 5.

[4] Let be a subgraph of the hypercube which is an AMP-amalgam of two ample isometric subgraphs and of . If is an isometric subgraph of , then is ample. Any ample partial cube can be obtained by AMP-amalgams from its facets.

Proof.

First we assert that any -cube of is contained either in or in . We proceed by induction on . Since is a separator, the assertion holds when . Suppose the assertion is true for all with and suppose that the -cube of contains two vertices and . By induction hypothesis, any facet of containing must be included in and any facet of containing must be included in . From this we conclude that all vertices of except and (which must be opposite in ) belong to . This is impossible since is ample.

By Proposition 1, we must show that any two -cubes of can be connected by a geodesic gallery. Since and are ample, this is true when and both belong to or to . By previous assertion we can suppose that and . By induction on we prove that and can be connected by a geodesic gallery containing an -cube of . If , then are vertices of separated by and we are done. So, let . Pick any , set , and let be the halfspaces of defined by . Let and be the intersections of and with the halfspaces. By induction hypothesis, can be connected by a geodesic gallery containing an -cube in and can be connected by a geodesic gallery containing an -cube in . Hence and . Since and are convex subgraphs of , and . Since is ample, the -cubes and can be connected in by a geodesic gallery. Since and , on this gallery we can find two consecutive -cubes and so that is an -cube of .

Since and are two -cubes of AMP , they can be connected in by a geodesic gallery . Analogously, and can be connected in by a geodesic gallery . We assert that the concatenation of the two galleries is a geodesic gallery between and , i.e., . Since , it suffices to show that and .

In each pick a vertex, say , such that each pair of vertices realizes the distance between the corresponding cubes. Then and and . Let and be two vertices of and , respectively, belonging to a shortest -path. Again, . Consequently, in we have and . Since is a partial cube, the interval is convex (Lemma 4), thus and belong to a common shortest path between and . Applying the same argument, we deduce that and belong to a common shortest path between and . Hence and , establishing that and . Consequently, , i.e., is a geodesic gallery. ∎

3.2.5. VC-dimension

The VC-dimension of OMs, COMs, and AMPs (all viewed as partial cubes) can be expressed in the following way:

Lemma 13.

If is an OM, then is the rank of . If is a COM, then is the largest VC-dimension of a face of . Finally, if is an AMP, then is the dimension of the