Geometry of Gaussoids

Transcription

1 Geometry of Gaussoids Bernd Sturmfels MPI Leipzig and UC Berkeley p 3 p 13 p 23 a 12 3 p 123 a 23 a 13 2 a 23 1 a 13 p 2 p 12 a 12 p p 1 Figure 1: With The vertices Tobias andboege, 2-faces ofalessio the 3-cube D Ali, are labeled and Thomas by the set of Kahle unknowns P [ A.

2 Matroids Figure The hypersimplex 2,4 is a regular octahedron. Its matroid subdivisions correspond to tropical lines in 3-space. A matroid is a combinatorial structure that encodes independence in linear algebra and geometry. The basis axioms reflect the ideal of homogeneousexample relations among We explain all the minors concepts ofina this rectangular section for thematrix simplest nontrivial case, r = 2 and m = 4. Fix the uniform matroid M = U 2,4. Here, ( ) ( ) 1 the Dressian equals the tropical Grassmannian, by the four-point-condition. Writing 0 for the space of phylogenetic 1 0 trees p23 p = on four taxa, 24 we have p 13 p 14 Dr M =Gr M =Dr(2, 4) = Gr(2, 4) =, A matroid is an assignment of 0 or to these minors so that the quadratic Plücker relations have a chance of vanishing: p 12 p 34 p 13 p 24 + p 14 p 23 = 0. We also like oriented matroids, positroids and valuated matroids. 34

3 Gaussoids A gaussoid is a combinatorial structure that encodes independence in probability and statistics. The gaussoid axioms reflect the ideal of homogeneous relations among the principal and almost-principal minors of a symmetric matrix ( ) ( ) p1 a = a 12 p 2 A gaussoid is an assignment of 0 or to these minors so that the quadratic Plücker relations have a chance of vanishing: p p 12 p 1 p 2 + a 2 12 = 0. Ditto: oriented gaussoids, positive gaussoids, valuated gaussoids. The gaussoid axioms were introduced in [R. Lněnička and F. Matúš: On Gaussian conditional independence structures, Kybernetika (2007)]

4 Principal and almost-principal minors A symmetric n n-matrix Σ has 2 n principal minors p I one for each subset I of [n] = {1, 2,..., n}. The matrix Σ has 2 n 2( n 2) almost-principal minors aij K. This is the subdeterminant of Σ with row indices {i} K and column indices {j} K, where i, j [n] and K [n]\{i, j}. Principal minors are in bijection with the vertices of the n-cube. Almost-principal minors are in bijection with the 2-faces of the n-cube. Σ = p 1 a 12 a 13 a 12 p 2 a 23 a 13 a 23 p 3 p 3 p 23 a 12 3 p 123 p 13 a 23 a 13 2 a 23 1 a 13 p 2 p 12 a 12 p p 1

5 Why Gauss? If Σ is positive definite then it is the covariance matrix of a Gaussian distribution on R n. In statistics: p I > 0 for all I [n]. Study n random variables X 1, X 2,..., X n, with the aim of learning how they are related. (Yes, data science) Almost-principal minors a ij K measure partial correlations. We have a ij K = 0 if and only if X i and X j are conditionally independent given X K. The inequalities a ij K > 0 and a ij K < 0 indicate whether conditional correlation is positive or negative.

6 Ideals, Varieties,... Write J n for the homogeneous prime ideal of relations among the principal and almost-principal minors of a symmetric n n-matrix. It lives in a polynomial ring R[p, a] with N = 2 n + 2 n 2( ) n 2 unknowns, and defines an irreducible subvariety of P N 1. Proposition The projective variety V (J n ) is a coordinate projection of the Lagrangian Grassmannian. They share dimension and degree: dim(v (J n )) = ( ) n+1 2 degree(v (J n )) = ( n+1 ) 2! 1 n 3 n 1 5 n 2 (2n 1) 1. The elimination ideal J n R[p] was studied by Holtz-St and Oeding. They found hyperdeterminantal relations of degree 4.

7 3-cube The ideal J 3 is generated by 21 quadrics. There are 9 quadrics associated with the facets of the 3-cube: S 200 (2, 0, 0) a pp 23 p 2 p 3 (0, 0, 0) 2a 23 a pp p 1 p 23 p 2 p 13 p 12 p 3 ( 2, 0, 0) a p 1p 123 p 12 p and two other such weight components There are 12 trinomials associated with the edge of the 3-cube: (1, 1, 0) a 13 a 23 + a 12 3 p a 12 p 3 S 110 (1, 1, 0) a 13 2 a 23 + a 12 3 p 2 a 12 p 23 ( 1, 1, 0) a 13 a a 12 3 p 1 a 12 p 13 ( 1, 1, 0) a 13 2 a a 12 3 p 12 a 12 p and two other such weight components The variety V (J 3 ) is the Lagrangian Grassmannian in P 13, which has dimension 6 and degree 16. It is arithmetically Gorenstein. Intersections with subspaces P 8 are canonical curves of genus 9.

8 3-cube Of most interest are the 12 edge trinomials: p 1 a 23 pa 23 1 a 12 a 13 p 2 a 13 pa 13 2 a 12 a 23 p 3 a 12 pa 12 3 a 23 a 13 p 12 a 13 p 1 a 13 2 a 12 a 23 1 p 12 a 23 p 2 a 23 1 a 12 a 13 2 p 13 a 12 p 1 a 12 3 a 13 a 23 1 p 13 a 23 p 3 a 23 1 a 13 a 12 3 p 23 a 12 p 2 a 12 3 a 23 a 13 2 p 23 a 13 p 3 a 13 2 a 23 a 12 3 p 123 a 12 p 12 a 12 3 a 23 1 a 13 2 p 123 a 13 p 13 a 13 2 a 23 1 a 12 3 p 123 a 23 p 23 a 23 1 a 12 3 a 13 2 p 3 p 13 p 23 a 12 3 p 123 a 23 a 13 2 a 23 1 a 13 p 2 p 12 a 12 p p 1

9 Gaussoid Axioms Let A be the set of ( n 2) 2 n 2 symbols a ij K. Following Lněnička and Matúš, a subset G of A is a gaussoid on [n] if it satisfies: 1. {a ij L, a ik jl } G implies {a ik L, a ij kl } G, 2. {a ij kl, a ik jl } G implies {a ij L, a ik L } G, 3. {a ij L, a ik L } G implies {a ij kl, a ik jl } G, 4. {a ij L, a ij kl } G implies ( a ik L G or a jk L G ). These axioms are known as 1. semigraphoid, 2. intersection, 3. converse to intersection, 4. weak transitivity.

10 Gaussoid Axioms Let A be the set of ( n 2) 2 n 2 symbols a ij K. Following Lněnička and Matúš, a subset G of A is a gaussoid on [n] if it satisfies: 1. {a ij L, a ik jl } G implies {a ik L, a ij kl } G, 2. {a ij kl, a ik jl } G implies {a ij L, a ik L } G, 3. {a ij L, a ik L } G implies {a ij kl, a ik jl } G, 4. {a ij L, a ij kl } G implies ( a ik L G or a jk L G ). These axioms are known as 1. semigraphoid, 2. intersection, 3. converse to intersection, 4. weak transitivity. Theorem The following are equivalent for a set G of 2-faces of the n-cube: (a) G is a gaussoid, i.e. the four axioms above are satisfied for G. (b) G is compatible with the quadratic edge trinomials in J n.

11 Duality and Minors Let G be any gaussoid on [n]. The dual of G is G = { a ij L : a ij K G and L = [n]\({i, j} K) }. Fix an element u [n]. The marginalization equals The conditioning equals G\u = { a ij K G : u {i, j} K }. G/u = { a ij K\{u} : a ij K G and u K }. Think of operations on sets of 2-faces of the n-cube. Proposition If G is a gaussoid on [n], and u [n], then G, G\u and G/u are gaussoids on [n]\{u}. The following duality relation holds: ( G\u ) = G /u and ( G/u ) = G \u. If G is realizable (with Σ positive definite) then so are G, G\u, G/u.

12 A Pinch of Representation Theory Fix the Lie group G = (SL 2 (C)) n. Write V i C 2 for the defining representation of the i-th factor. The irreducible G-modules are S d1d 2 d n = n Sym di (V i ), Proposition G acts on the space W pr spanned by the principal minors and the spaces W ij ap spanned by almost-principal minors. As G-modules, i=1 W pr n i=1v i and W ij ap k [n]\{i,j} V k for 1 i < j n. This defines the G-action and Z n -grading on our polynomial ring C[p, a]. The formal character of C[p, a] 1 = W pr i,j W ij ap is the sum of weights: n i=1 (x i + x 1 i ) + 1 i<j n k [n]\{i,j} (x k + x 1 k )

13 Commutative Algebra The number of linearly independent quadrics in the ideal J n equals ( ) n n 3 ( ) n n 2 ( ) n 3 n k (n k)(n k 1) + 2 k3 n 2k 2 k 2k k=0 Derived via the lowering and raising operators in the Lie algebra g. Conjecture These quadrics generate J n. Proposition The number of face trinomials and edge trinomials equals ( ) ( ) n n 2 n n 3 = 2 n 3 n(n 1)(2n 3). 2 3 k=2 These trinomials generate the image of J n in C[p, a ± ].

14 Page 1 of 1 4-cube p3 p23 a12 3 p13 p123 a23 a13 2 a23 1 a13 p12 p2 a12 p p1 There are 16 principal and 24 almost principal minors. They span Figure 1: The vertices and 2-faces of the 3-cube are labeled by the set of unknowns P [ A. Example 2.2 (n = 3). The polynomial ring R[P [ A] has eight unknowns pi and six unknowns aij K. These are identified respectively with the vertices and facets of the 3-cube. This is shown in Figure 1. The variety V (J3) hasdimension6anddegree6!/( ) = 16 in P 13. The ideal J3 is minimally generated by the 21 quadrics shown in the following tables: 2 3 (2, 0, 0) a pp23 p2p3 S200 4 (0, 0, 0) 2a23a pp123 + p1p23 p2p13 p12p3 5 ( 2, 0, 0) a p1p123 p12p (0, 2, 0) a pp13 p1p3 S020 4 (0, 0, 0) 2a13a pp123 + p2p13 p1p23 p12p3 5 (0, 2, 0) a p2p123 p12p23 2 (0, 0, 2) a pp12 p1p2 3 S002 4 (0, 0, 0) 2a12a pp123 + p3p12 p1p23 p13p2 5 (0, 0, 2 2) a p3p123 p13p23 3 (1, 1, 0) a13a23 + a12 3p a12p3 S ( 1, 1, 0) a13a a12 3p1 a12p13 ( 1, 2 1, 0) a13 2a a12 3p12 a12p123 3 (1, 0, 1) a12a23 + a13 2p a13p2 S101 6 (1, 0, 1) a12 3a23 + a13 2p3 a13p ( 1, 0, 1) a12a a13 2p1 a13p12 ( 1, 0, 2 1) a12 3a a13 2p13 a13p123 3 (0, 1, 1) a12a13 + a23 1p a23p1 S011 6 (0, 1, 1) a12 3a13 + a23 1p3 a23p (0, 1, 1) a12a a23 1p2 a23p12 (0, 1, 1) a12 3a a23 1p23 a23p123 C[p, a] 1 = S 1111 S 1100 S 1010 S 1001 S 0110 S 0101 S The space of quadrics has dimension 820. As G-module, C[a, p] 2 S 2222 S 2211 S 2121 S 2112 S 1221 S 1212 S S S S S S S S S S S S S S S S S S S S S S S S S S S The 226-dimensional submodule (J 4 ) 2 of quadrics in our ideal equals S 2200 S 2020 S 2002 S 0220 S 0202 S 0022 S 2110 S 2101 S 2011 S 1210 S 1201 S 0211 S 1120 S 1021 S 0121 S 1102 S 1012 S 0112 S 1100 S 1010 S 1001 S 0110 S 0101 S S Of these, 120 are trinomials: 96 edge trinomials and 24 face trinomials. 3

15 Enumeration of Gaussoids Theorem The number of gaussoids for n = 3, 4, 5 equals: n all gaussoids orbits for S n Z/2Z S n (Z/2Z) n S n , 212, , , , 981 For n = 3, all 11 gaussoids are realizable: {}, {a 12 }, {a 13 }, {a 23 }, {a 12 3 }, {a 13 2 }, {a 23 1 }, {a 12, a 12 3, a 13, a 13 2 }, {a 12, a 12 3, a 23, a 23 1 }, {a 13, a 13 2, a 23, a 23 1 }, {a 12, a 12 3, a 13, a 13 2, a 23, a 23 1 }.

16 Enumeration of Gaussoids Theorem The number of gaussoids for n = 3, 4, 5 equals: n all gaussoids orbits for S n Z/2Z S n (Z/2Z) n S n , 212, , , , 981 For n = 3, all 11 gaussoids are realizable: {}, {a 12 }, {a 13 }, {a 23 }, {a 12 3 }, {a 13 2 }, {a 23 1 }, {a 12, a 12 3, a 13, a 13 2 }, {a 12, a 12 3, a 23, a 23 1 }, {a 13, a 13 2, a 23, a 23 1 }, {a 12, a 12 3, a 13, a 13 2, a 23, a 23 1 }. For n = 4, five of the 42 gaussoid classes are non-realizable. For instance, G = {a 12 3, a 13 4, a 14 2 } is not realizable. Real Nullstellensatz certificate: a 14 ( a 2 34 p 2 p 4 p 23 + a 2 23 a2 34 p 24 + p 2 2 p 3p 4 p 34 ) (a 23 a 24 a 34 + p 2 p 3 p 4 )(a 24 p 4 a a 24 a 23 a p 3 p 4 a 14 2 ) J 4.

17 SAT Solvers Current software for the satisfiability problem is very impressive, and useful for enumerating combinatorial structures like gaussoids. The input is a Boolean formula in conjunctive normal form (CNF). One can specify one of the following three output options: SAT: Is the formula satisfiable? #SAT: How many satisfying assignments are there? AllSAT: Enumerate all satisfying assignments. We found the 60, 212, 776 gaussoids for n = 5 in about one hour using Thurley s software bdd minisat all. The input was a SAT formulation of the gaussoid axioms using 1680 clauses in the CNF. We then analyzed the output with respect to the symmetry groups.

18 Oriented gaussoids An oriented gaussoid is a map A {0, ±1} such that, for each edge trinomial, after setting each p I to +1 and each a ij K to its image, the set of signs of terms is {0} or { 1, +1} or { 1, 0, +1}. Analogous to oriented matroids. A positive gaussoid is an assignment A {0, +1} with the same compatibility requirement. Analogous to positroids.

19 Oriented gaussoids An oriented gaussoid is a map A {0, ±1} such that, for each edge trinomial, after setting each p I to +1 and each a ij K to its image, the set of signs of terms is {0} or { 1, +1} or { 1, 0, +1}. Analogous to oriented matroids. A positive gaussoid is an assignment A {0, +1} with the same compatibility requirement. Analogous to positroids. Example Let n = 3. Each singleton gaussoid, like G = {a 12 } or {a 12 3 } supports four oriented gaussoids, related by reorientation. We display these 24 = 6 4 oriented gaussoids by listing the six signs for A in the order a 12, a 13, a 23, a 12 3, a 13 2, a 23 1 :

20 3-Cube and Beyond Proposition For n=3 there are 51 oriented gaussoids in seven symmetry classes. All are realizable. This includes 20 uniform gaussoids A {±1}. The following table exhibits the seven classes. The first column gives a covariance matrix Σ that realizes the first oriented gaussoid in the class: (p 1,p 2,p 3, a 12,a 13,a 23 ) Symmetry class of oriented gaussoids (2, 2, 2, 1, 1, 1) , + +, + +, + + (3, 5, 1, 1, 1, 2) , +, ++ +,..., + (6, 9, 6, 1, 1, 7), ++ ++, ++ ++, (4, 3, 3, 2, 2, 1) , ,... previous page (2, 2, 2, 0, 1, 1) 0, ,... previous page (3, 2, 2, 0, 0, 1) , 00 00, 00 00,..., (1, 1, 1, 0, 0, 0)

21 3-Cube and Beyond Proposition For n=3 there are 51 oriented gaussoids in seven symmetry classes. All are realizable. This includes 20 uniform gaussoids A {±1}. The following table exhibits the seven classes. The first column gives a covariance matrix Σ that realizes the first oriented gaussoid in the class: (p 1,p 2,p 3, a 12,a 13,a 23 ) Symmetry class of oriented gaussoids (2, 2, 2, 1, 1, 1) , + +, + +, + + (3, 5, 1, 1, 1, 2) , +, ++ +,..., + (6, 9, 6, 1, 1, 7), ++ ++, ++ ++, (4, 3, 3, 2, 2, 1) , ,... previous page (2, 2, 2, 0, 1, 1) 0, ,... previous page (3, 2, 2, 0, 0, 1) , 00 00, 00 00,..., (1, 1, 1, 0, 0, 0) Theorem The number of oriented gaussoids is 34,873 for n = 4, and it is for n = 5. Among these, are uniform.

22 From Positroids to Statistics Positroids are oriented matroids whose bases are positive. These are important in representation theory and algebraic combinatorics, and they have desirable topological properties. Positive gaussoids correspond to distributions that are of current interest in statistics: S.Fallat, S.Lauritzen, K.Sadeghi, C.Uhler, N.Wermuth and P.Zwiernik: Total positivity in Markov structures, Annals of Statistics 45 (2017) F. Mohammadi, C. Uhler, C. Wang and J. Yu: Generalized permutohedra from probabilistic graphical models, arxiv: S. Lauritzen, C. Uhler and P. Zwiernik: Maximum likelihood estimation in Gaussian models under total positivity, arxiv:

23 From Positroids to Statistics Positroids are oriented matroids whose bases are positive. These are important in representation theory and algebraic combinatorics, and they have desirable topological properties. Positive gaussoids correspond to distributions that are of current interest in statistics: S.Fallat, S.Lauritzen, K.Sadeghi, C.Uhler, N.Wermuth and P.Zwiernik: Total positivity in Markov structures, Annals of Statistics 45 (2017) F. Mohammadi, C. Uhler, C. Wang and J. Yu: Generalized permutohedra from probabilistic graphical models, arxiv: S. Lauritzen, C. Uhler and P. Zwiernik: Maximum likelihood estimation in Gaussian models under total positivity, arxiv: Ardila, Rincón and Williams (2017) proved a 1987 conjecture of Da Silva by showing that all positroids are realizable. We derive the analogue for gaussoids: all positive gaussoids are realizable and their realization spaces are very nice.

24 Positive Gaussoids are Graphical Every graph Γ = ([n], E) defines a gaussoid G Γ via CI statements that hold for the graphical model Γ. Here, a ij K lies in G Γ iff every path from i to j in Γ passes through K. Thus a ij G Γ when i and j are disconnected in Γ, and a ij [n]\{i,j} G Γ when {i, j} E. Theorem For n 2, there are precisely 2 (n 2) positive gaussoids. All are realizable from graphs as above. The space of covariance matrices Σ that realize G Γ is homeomorphic to a ball of dimension E + n. The concentration matrices Σ 1 are M-matrices with support Γ. [S. Karlin and Y. Rinott: M-matrices as covariance matrices of multinormal distributions, Linear Algebra Appl. (1983)] Positive gaussoids satisfy the axiomatic requirements in [K. Sadeghi: Faithfulness of probability distributions and graphs, arxiv:1701..]

25 Conclusion Matroids are cool. And so are gaussoids. Positivity is crucial in algebraic combinatorics. And in statistics. On this journey, let the quadratic equations be your guide. Hitch a fast ride using SAT solvers and representation theory. p 1 a 23 pa 23 1 a 12 a 13, p 2 a 13 pa 13 2 a 12 a 23, p 3 a 12 pa 12 3 a 23 a 13, p 12 a 13 p 1 a 13 2 a 12 a 23 1, p 12 a 23 p 2 a 23 1 a 12 a 13 2,...

26 Conclusion Matroids are cool. And so are gaussoids. Positivity is crucial in algebraic combinatorics. And in statistics. On this journey, let the quadratic equations be your guide. Hitch a fast ride using SAT solvers and representation theory. p 1 a 23 pa 23 1 a 12 a 13, p 2 a 13 pa 13 2 a 12 a 23, p 3 a 12 pa 12 3 a 23 a 13, p 12 a 13 p 1 a 13 2 a 12 a 23 1, p 12 a 23 p 2 a 23 1 a 12 a 13 2,... p 3 p 13 p 23 a 12 3 p 123 a 23 a 13 2 a 23 1 a 13 p 2 p 12 a 12 p p 1 Thank You Figure 1: The vertices and 2-faces of the 3-cube are labeled by the set of unknowns P [ A. Stay tuned for valuated gaussoids via tropical Lagrangian Grassmannian. Example 2.2 (n = 3). The polynomial ring R[P [ A] has eight unknowns p I and six