Convex rank tests Anne Shiu Texas A&M University CombinaTexas - PowerPoint PPT Presentation

Convex rank tests Anne Shiu Texas A&M University CombinaTexas 8 May 2016

From Algebraic Systems Biology: A Case Study for the Wnt Pathway (Elizabeth Gross, Heather Harrington, Zvi Rosen, Bernd Sturmfels 2016).

Outline of talk ◮ Introduction ◮ Main Result: convex rank tests = semigraphoids ◮ 2 counterexamples ◮ Application to biology

Outline of talk ◮ Introduction ◮ Main Result: convex rank tests = semigraphoids ◮ 2 counterexamples ◮ Application to biology Joint work with Raymond Hemmecke, Jason Morton, Lior Pachter, Bernd Sturmfels, and Oliver Wienand.

Introduction

Preliminary definitions ◮ A fan in R n is a finite collection F of polyhedral cones such that: ◮ if C ∈ F and C ′ is a face of C , then C ′ ∈ F , and ◮ if C, C ′ ∈ F , then C ∩ C ′ is a face of C . ◮ The S n -arrangement (the braid arrangement ) is the arrangement of hyperplanes { x i = x j } in R n . ◮ Example: the fan associated to the S 3 - arrangement has 6 maximal cones. x 1 = x 3 x 1 = x 2

What is a convex rank test? ◮ A rank test is a partition of S n . ◮ A convex rank test is a partition of S n defined by a fan that coarsens the S n -arrangement. ◮ Example: the following convex rank test partitions S 3 into 4 classes. 123 132 312 ( x 3 > x 1 > x 2 ) 213 231 321

A non-convex rank test ◮ This partition of S 3 into 4 classes is not a convex rank test. 123 132 213 312 231 321 ◮ Remark : a convex rank test is determined by the walls removed from the S n -arrangement.

Label walls by conditional-independence statements 123 132 1 ⊥ ⊥ 3 |∅ 213 312 1 ⊥ ⊥ 3 |{ 2 } 231 321 ◮ Two maximal cones of the S n -fan, labeled by permutations δ and δ ′ in S n , share a wall if δ and δ ′ differ by an adjacent transposition : there exists an index k such that δ k = δ ′ k +1 , δ k +1 = δ ′ k , and δ i = δ ′ i for i � = k, k + 1. ◮ Label wall { δ, δ ′ } by the conditional-independence (CI) statement: δ k ⊥ ⊥ δ k +1 | { δ 1 , . . . , δ k − 1 } .

Conditional independence Consider a collection of n random variables indexed by [ n ] . [1 ⊥ ⊥ 2 |∅ ] [1 ⊥ ⊥ 3 |∅ ] [2 ⊥ ⊥ 3 |∅ ] [2 ⊥ ⊥ 3 | 1] [1 ⊥ ⊥ 3 | 2] [1 ⊥ ⊥ 2 | 3] [1 ⊥ ⊥ 4 |∅ ] [2 ⊥ ⊥ 4 |∅ ] [3 ⊥ ⊥ 4 |∅ ] [1 ⊥ ⊥ 2 | 4] [1 ⊥ ⊥ 3 | 4] [2 ⊥ ⊥ 3 | 4] [2 ⊥ ⊥ 4 | 1] [3 ⊥ ⊥ 4 | 1] [1 ⊥ ⊥ 4 | 2] [3 ⊥ ⊥ 4 | 2] [1 ⊥ ⊥ 4 | 3] [2 ⊥ ⊥ 4 | 3] [1 ⊥ ⊥ 2 | 34] [1 ⊥ ⊥ 3 | 24] [1 ⊥ ⊥ 4 | 23] [2 ⊥ ⊥ 3 | 14] [2 ⊥ ⊥ 4 | 13] [3 ⊥ ⊥ 4 | 12] [1 ⊥ ⊥ 5 |∅ ] [2 ⊥ ⊥ 5 |∅ ] . . . [4 ⊥ ⊥ 5 | 123] . . . The symbol [ i ⊥ ⊥ j | K ] represents the statement, “the random variables i and j are conditionally independent given the joint random variable K .”

Semigraphoids ◮ (definition #1) A set M of CI statements on [ n ] is a semigraphoid if the following axiom holds 1 : ( SG ) If [ i ⊥ ⊥ j | K ∪ ℓ ] and [ i ⊥ ⊥ ℓ | K ] are in M then also [ i ⊥ ⊥ j | K ] and [ i ⊥ ⊥ ℓ | K ∪ j ] are in M . 1 Probabilistic Conditional Independence Structures , Studen´ y 2005

Semigraphoids ◮ (definition #1) A set M of CI statements on [ n ] is a semigraphoid if the following axiom holds 1 : ( SG ) If [ i ⊥ ⊥ j | K ∪ ℓ ] and [ i ⊥ ⊥ ℓ | K ] are in M then also [ i ⊥ ⊥ j | K ] and [ i ⊥ ⊥ ℓ | K ∪ j ] are in M . ◮ Example: ( SG ) If [1 ⊥ ⊥ 2 | 3] and [1 ⊥ ⊥ 3 |∅ ] are in M , then also [1 ⊥ ⊥ 2 |∅ ] and [1 ⊥ ⊥ 3 | 2] are in M . ◮ So, M = { [1 ⊥ ⊥ 3 |∅ ] , [1 ⊥ ⊥ 2 | 3] } is not a semigraphoid. 123 132 1 ⊥ ⊥ 3 |{∅} 213 312 1 ⊥ ⊥ 2 |{ 3 } 231 321 1 Probabilistic Conditional Independence Structures , Studen´ y 2005

Main result

Main Theorem ◮ A convex rank test F is characterized by the collection of walls { δ, δ ′ } that are removed from the S n -arrangement. Let M F denote the CI statements that label those walls. ◮ Main theorem: The map F �→ M F is a bijection between convex rank tests and semigraphoids. ◮ The following convex rank test corresponds to the semigraphoid M = { 1 ⊥ ⊥ 3 |∅ , 1 ⊥ ⊥ 3 | 2 } . 123 132 1 ⊥ ⊥ 3 |∅ 213 312 1 ⊥ ⊥ 3 |{ 2 } 231 321

Restating the Main result via the permutohedron

The Permutohedron ◮ The fan of the S n -arrangement is the normal fan of the permutohedron P n (the convex hull of the vectors ( ρ 1 , . . . , ρ n ), where ρ is in S n ). 123 • 132 • 1 ⊥ ⊥ 3 |∅ 213 • • 312 1 ⊥ ⊥ 3 |{ 2 } 231 • • 321 ◮ The edges of the permutohedron correspond to walls of the S n -arrangement.

The permutohedron P 4 2 ⊥ ⊥ 3 | 14 • • • • • • • • • • • • • 1432 • • 1423 4312 • 4213 ◦ 1342 • • 1243 4321 ◦ ◦ 4231 3412 • 2413 ◦ • • 3142 1234 • 2143 1324 • 3421 • ◦ 2431 2134 • 3124 • 3241 • ◦ 2341 • • 3214 2314 The 2-d faces of P n are squares and hexagons.

Square and Hexagon Axioms Lemma: A set M of edges of the permutohedron P n is a semigraphoid if and only if M satisfies the following two axioms: ◮ Square axiom: Whenever an edge of a square is in M , then the opposite edge is also in M . ◮ Hexagon axiom: When two adjacent edges of a hexagon are in M , then the two opposite edges are also in M . 123 • 132 • 213 • • 312 231 • • 321 Main theorem, restated. Coarsenings of the S n -fan are equivalent to subsets of edges of P n that satisfy the Square and Hexagon axioms . Generalization to other Coxeter arrangemts. Coarsenings = subsets of edges with the polygon property . (Nathan Reading 2012).

Hexagon axiom illustrated Consider M = { 1 ⊥ ⊥ 3 |∅ , 1 ⊥ ⊥ 2 |{ 3 }} (again). It is not a convex rank test, because it violates the Hexagon axiom: 123 • 132 • 1 ⊥ ⊥ 3 |∅ 213 • • 312 1 ⊥ ⊥ 2 |{ 3 } 231 • • 321

Main theorem illustrated 4132 • 4123 • • • • • • • • • • • • 1432 • • 1423 • • • • • • • • • • • 4312 • 4213 ◦ • • • • 1342 • • • 1243 • • 4321 ◦ • • ◦ • 4231 • • 3412 • 2413 ◦ • • • • • • • • • • • 3142 • 1234 • • 2143 1324 • • • • • • • 3421 • ◦ • • 2431 • • 2134 • • 3124 • • 3241 • ◦ 2341 • • • • • • • • • • • • • • 3214 2314 f = (16 , 24 , 10)

2 counterexamples

Semigraphoids: another definition ◮ Each CI statement defines a linear form in 2 n unknowns h I for I ⊆ [ n ]: [ i ⊥ ⊥ j | K ] �→ − h ijK + h iK + h jK − h K . ◮ Non-negativity of these linear forms defines the (2 n − n − 1)-dimensional submodular cone in R 2 n . ◮ The linear relations among the forms are spanned by entropy equations: [ i ⊥ ⊥ j | K ∪ ℓ ] + [ i ⊥ ⊥ ℓ | K ] = [ i ⊥ ⊥ j | K ] + [ i ⊥ ⊥ ℓ | K ∪ j ] .

Semigraphoids: another definition ◮ Each CI statement defines a linear form in 2 n unknowns h I for I ⊆ [ n ]: [ i ⊥ ⊥ j | K ] �→ − h ijK + h iK + h jK − h K . ◮ Non-negativity of these linear forms defines the (2 n − n − 1)-dimensional submodular cone in R 2 n . ◮ The linear relations among the forms are spanned by entropy equations: [ i ⊥ ⊥ j | K ∪ ℓ ] + [ i ⊥ ⊥ ℓ | K ] = [ i ⊥ ⊥ j | K ] + [ i ⊥ ⊥ ℓ | K ∪ j ] . ◮ (definition #4) A semigraphoid M specifies the possible zeros for a non-negative solution of the entropy equations. ◮ A semigraphoid M is submodular if it is the set of actual zeros of a point in the submodular cone.

Question 1 ◮ Postnikov, Reiner and Williams (2006) asked: Is every simplicial fan which coarsens the S n -fan the normal fan of convex polytope? ◮ Facts. A convex rank test F is the normal fan of a polytope if and only if the semigraphoid M F is submodular. This polytope is a generalized permutohedron . It is simple iff F is simplicial iff the posets on [ n ] are trees. ◮ The answer to the PRW question is no for n = 4: Proposition. This is simplicial, but not submodular: • • • • • • • • • • • • • • • ◦ • • • • • • ◦ ◦ • • • • ◦ • • • • • • • • • • • • ◦ • • • • • • • ◦ • • • • • • • •

Proof: simplicial This simple polytope looks like a generalized permutohedron... 41 23 14 23 32 32 • • • • • • • • • • • • • • • 1342 4213 • ◦ ◦ ◦ ◦ ◦ 12 ◦ • 21 43 • ◦ • ◦ • ◦ • • ◦ • • ◦ 2413 1324 • ◦ ◦ ◦ ◦ • 4231 • • • • • 12 21 34 ◦ ◦ • ◦ ◦ 2431 ◦ ◦ • ◦ 23 14 41 M F = { [2 ⊥ ⊥ 3 | 14] , [1 ⊥ ⊥ 4 | 23] , [1 ⊥ ⊥ 2 |∅ ] , [3 ⊥ ⊥ 4 |∅ ] } .