Normal and Unimodular Hierarchical Models Daniel Irving Bernstein and Seth Sullivant North Carolina State University dibernst@ncsu.edu http://www4.ncsu.edu/~dibernst/ http://arxiv.org/abs/1502.06131 http://arxiv.org/abs/1508.05461 October 3, 2015 Daniel Irving Bernstein Normal and Unimodular Hierarchical Models October 3, 2015 1 / 19
Example Let T be the following 3 × 2 × 2 table front back 1 1 2 1 2 0 1 3 3 1 0 2 If we sum entries going down, we get the 2-way margin below. If we sum entries going left and back, we get the 1-way margin below. 5 � 3 � 6 6 6 2 6 We are interested in the matrix that maps tables to margins Daniel Irving Bernstein Normal and Unimodular Hierarchical Models October 3, 2015 2 / 19
Main Definition d = ( d 1 , d 2 , . . . , d n ) is an integer vector, d i ≥ 2 C denotes a simplicial complex on [ n ] facet( C ) denotes the inclusion-maximal faces of C Definition Let A C , d be the matrix defined as follows: Columns are indexed by elements of � n i =1 [ d i ] Rows are indexed by � � j ∈ F [ d j ] F ∈ facet( C ) Entry in row ( F , ( j 1 , . . . , j k )) and column ( i 1 , . . . , i n ) is 1 if i | F = ( j 1 , . . . , j k ) All other entries are 0 Daniel Irving Bernstein Normal and Unimodular Hierarchical Models October 3, 2015 3 / 19
Example Let n = 3 with d 1 = 3 , d 2 = 2 , d 3 = 2 1 2 3 Let C be the complex Then A C , d is the following matrix: 1 1 1 1 2 2 2 2 3 3 3 3 1 2 2 1 1 2 2 1 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 { 1 } , 1 1 1 1 1 0 0 0 0 0 0 0 0 { 1 } , 2 0 0 0 0 1 1 1 1 0 0 0 0 { 1 } , 3 0 0 0 0 0 0 0 0 1 1 1 1 { 2 , 3 } , 11 1 0 0 0 1 0 0 0 1 0 0 0 { 2 , 3 } , 12 0 1 0 0 0 1 0 0 0 1 0 0 { 2 , 3 } , 21 0 0 1 0 0 0 1 0 0 0 1 0 { 2 , 3 } , 22 0 0 0 1 0 0 0 1 0 0 0 1 Daniel Irving Bernstein Normal and Unimodular Hierarchical Models October 3, 2015 4 / 19
Motivating Question Definition (Unimodularity) Assume A ∈ Z d × n has full row rank. We say that A is unimodular if all nonsingular d × d submatrices have determinant ± 1. Definition (Normality) We say that A ∈ Z d × n is normal if Z A ∩ R ≥ 0 A = N A . This is a weaker condition than unimodularity. Question When is A C , d unimodular? When is it normal? Observation If A C , d is unimodular/normal, then so is A C , (2 ,..., 2) . Daniel Irving Bernstein Normal and Unimodular Hierarchical Models October 3, 2015 5 / 19
Our Results Our results include: Necessary and sufficient conditions on C guaranteeing unimodularity of A C , 2 Progress towards a similar classification for normal A C , 2 Note We abuse language and say that a simplicial complex C is unimodular/normal to mean that A C , (2 ,..., 2) is unimodular/normal. Applications include: Integer programming Disclosure limitation Compute Markov basis via toric fiber product (Rauh-Sullivant 2014) Daniel Irving Bernstein Normal and Unimodular Hierarchical Models October 3, 2015 6 / 19
Unimodularity-Preserving Operations Definition (Adding a cone vertex) If C is a simplicial complex on [ n ], define cone( C ) to be the complex on [ n + 1] with facets facet(cone( C )) = { F ∪ { n + 1 } : F ∈ facet( C ) } . Definition (Adding a ghost vertex) If C is a simplicial complex on [ n ], define G ( C ) to be the simplicial complex on [ n + 1] that has exactly the same faces as C . Definition (Alexander Duality) If C is a simplicial complex on [ n ], then the Alexander dual complex C ∗ is the simplicial complex on [ n ] with facets facet( C ∗ ) = { [ n ] \ S : S is a minimal non-face of C} . Daniel Irving Bernstein Normal and Unimodular Hierarchical Models October 3, 2015 7 / 19
Unimodularity: Constructive Classification Definition We say that a simplicial complex C is nuclear if it satisfies one of the following: 1 C = ∆ k for some k ≥ − 2 (i.e. a simplex) 2 C = ∆ m ⊔ ∆ n (i.e. a disjoint union of simplices) 3 C = cone( D ) where D is nuclear 4 C = G ( D ) where D is nuclear 5 C is the Alexander dual of a nuclear complex. Theorem (B-Sullivant 2015) The matrix A C is unimodular if and only if C is nuclear. Daniel Irving Bernstein Normal and Unimodular Hierarchical Models October 3, 2015 8 / 19
Simplicial Complex Minors Definition (Deletion) Let C be a simplicial complex on [ n ]. Let v ∈ [ n ] be a vertex of C . Then C \ v denotes the induced simplicial complex on [ n ] \ { v } . Definition (Link) Let C be a simplicial complex on [ n ]. Let v ∈ [ n ] be a vertex of C . Then link v ( C ) denotes the simplicial complex on [ n ] \ { v } with facets facet(link v ( C )) = { F \ { v } : F is a facet of C with v ∈ F } . Definition (Simplicial Complex Minor) Let C , D be simplicial complexes. If D can be obtained from C by taking links of vertices and deleting vertices, then we say that D is a minor of C . Daniel Irving Bernstein Normal and Unimodular Hierarchical Models October 3, 2015 9 / 19
Unimodularity: Excluded Minor Classification Theorem (B-Sullivant 2015) The matrix A C is unimodular if and only if C has no simplicial complex minors isomorphic to any of the following ∂ ∆ k ⊔ { v } , the disjoint union of the boundary of a simplex and an isolated vertex O 6 , the boundary complex of an octahedron, or its Alexander dual O ∗ 6 The four simplicial complexes shown below Daniel Irving Bernstein Normal and Unimodular Hierarchical Models October 3, 2015 10 / 19
Sketch of Proof C nuclear = ⇒ C unimodular Simplices are unimodular A disjoint union of two simplices is unimodular Adding cone and ghost vertices and taking duals preserves unimodularity C unimodular = ⇒ C avoids forbidden minors The forbidden minors are not unimodular Taking minors preserves unimodularity C avoids forbidden minors = ⇒ C nuclear If C avoids the forbidden minors but has a 4-cycle, then it must be an iterated cone over the 4-cycle. This is nuclear. So focus on 4-cycle-free complexes. Then the 1-skeleton is either a complete graph, or two complete graphs glued along a clique. Complex induction argument based on the link of a vertex of C . Daniel Irving Bernstein Normal and Unimodular Hierarchical Models October 3, 2015 11 / 19
Next Steps - Unimodularity Question Given a simplicial complex C on [ n ] and an integer vector d = ( d 1 , . . . , d n ) with d i ≥ 2, is A C , d unimodular? Corollary (B-Sullivant 2015) If A C , d is unimodular, then C is nuclear. Question Let C and d be specified by the figure below. For which values of p and q is A C , d unimodular? 3 p 2 q Daniel Irving Bernstein Normal and Unimodular Hierarchical Models October 3, 2015 12 / 19
Known Classification Results - Normality Theorem (Sullivant 2010) If C is a graph, then A C , 2 is normal if and only if C is free of K 4 -minors. Theorem (Bruns, Hemmecke, Hibi, Ichimc, Ohsugi, Kpped, Sgera 2007-2011) Let C be a complex whose facets are all m − 1 element subsets of [ m ] . Then A C , d is normal in precisely the following situations up to symmetry: 1 At most two of the d v are greater than two 2 m = 3 and d = (3 , 3 , a ) for any a ∈ N 3 m = 3 and d = (3 , 4 , 4) , (3 , 4 , 5) or (3 , 5 , 5) . Theorem (Rauh-Sullivant 2014) Let C be the four-cycle graph. Then A C , d is normal if d = (2 , a , 2 , b ) or d = (2 , a , 3 , b ) with a , b , ∈ N . Daniel Irving Bernstein Normal and Unimodular Hierarchical Models October 3, 2015 13 / 19
Corollary of Unimodular Classification Definition Let C be a simplicial complex on [ n ]. We say a facet of C that has n − 1 vertices is called a big facet . Proposition If C is a complex with a big facet, then C is normal if and only if unimodular. So our classification result on unimodular C immediately gives a classification of the normal C when C has a big facet. Daniel Irving Bernstein Normal and Unimodular Hierarchical Models October 3, 2015 14 / 19
Normality Preserving Operations Theorem (Sullivant 2010) Normality of A C , d is preserved under the following operations on the simplicial complex 1 Deleting vertices 2 Contracting edges 3 Gluing two simplicial complexes along a common face 4 Adding or removing a cone or ghost vertex. Theorem (B-Sullivant 2015) Normality of A C , d is preserved when taking links of vertices of C . Daniel Irving Bernstein Normal and Unimodular Hierarchical Models October 3, 2015 15 / 19
Minimally Non-Normal Simplicial Complexes Question Which simplicial complexes are minimally non-normal with respect to the operations of deleting vertices, contracting edges, gluing two complexes along a facet, removing cone and ghost vertices, and taking links of vertices? Computational method: All simplicial complexes on 3 or fewer vertices are normal Choose two normal simplicial complexes C , D on n − 1 vertices. Create simplicial complex C ′ on n vertices by attaching a new vertex v to C such that link v ( C ′ ) = D See if (non)normality of C ′ can be certified by reducing to a smaller complex via our normality-preserving operations If not, check normality of C ′ using Normaliz. If non-normal, then minimally non-normal Daniel Irving Bernstein Normal and Unimodular Hierarchical Models October 3, 2015 16 / 19
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