ALGEBRA: From Linear to Non-Linear. Bernd Sturmfels University of California at Berkeley

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1 ALGEBRA: From Linear to Non-Linear Bernd Sturmfels University of California at Berkeley John von Neumann Lecture, SIAM Annual Meeting, Pittsburgh, July 13, 2010

2 Undergraduate Linear Algebra All undergraduate students learn about Gaussian elimination, a general method for solving linear systems of algebraic equations: Input: x + 2y + 3z = 5 7x + 11y + 13z = 17 19x + 23y + 29z = 31

3 Undergraduate Linear Algebra All undergraduate students learn about Gaussian elimination, a general method for solving linear systems of algebraic equations: Input: Output: x + 2y + 3z = 5 7x + 11y + 13z = 17 19x + 23y + 29z = 31 x = 35/18 y = 2/9 z = 13/6 Solving very large linear systems is central to applied mathematics.

4 Undergraduate Non-Linear Algebra Lucky undergraduate students also learn about Gröbner bases, a general method for non-linear systems of algebraic equations: Input: x 2 + y 2 + z 2 = 2 x 3 + y 3 + z 3 = 3 x 4 + y 4 + z 4 = 4

5 Undergraduate Non-Linear Algebra Lucky undergraduate students also learn about Gröbner bases, a general method for non-linear systems of algebraic equations: Input: x 2 + y 2 + z 2 = 2 x 3 + y 3 + z 3 = 3 x 4 + y 4 + z 4 = 4 Output: 3z 12 12z 10 12z 9 +12z 8 +72z 7 66z 6 12z 4 +12z 3 1 = 0 4y 2 + (36z z 10 69z 9 252z 8 216z z 6 +72z 5 12z 4 99z 3 +10z+3) y + 36z z 10 72z 9 234z 8 192z z 6 48z 5 +96z 4 96z 3 +10z 2 +8 = 0 4x + 4y + 36z z 10 69z 9 252z 8 216z z 6 +72z 5 12z 4 99z 3 +10z+3 = 0 Non-linear equations can be intimidating, but they are important.

6 This Lecture What I shall speak about: Chemical Reaction Networks Convex Algebraic Geometry Tropical Mathematics What others will speak about: Numerical Algebraic Geometry Algebraic Methods in Discrete Optimization Multilinear Algebra and Tensors Algebraic Statistics

7 Topic 1: CHEMICAL REACTION NETWORKS Chemical species A, B, H, O,... can form complexes that react: A + B 2H + O We write complexes in multiplicative notation: AB H 2 O A chemical reaction network is a collection of such reactions. Associated with each reaction is a rate constant κ R >0. The concentrations c (t) of the species are functions of time t. Relevant authors include F. Horn, R. Jackson, M. Feinberg, E. Sontag, D. Anderson, G. Gnacadja, J. Gunawardena, C. Conradi, J. Stelling, G. Craciun, W. Helton, P. De Leenheer, A. Shiu,... My point: Reaction network theory is important for systems biology, and non-linear algebra can make useful contributions. R. Laubenbacher and BSt: Computer algebra in systems biology, Amer. Math. Monthly 116 (2009)

8 Networks and their Siphons Suppose there are s species that form n complexes. These correspond to monomials c y 1,c y 2,...,c yn in the unknown concentrations c 1,...,c s. Our chemical reaction network is a directed graph G whose nodes are labeled by these monomials. A siphon is a subset S of species such that all reactions in a component of G are shut off when c i = 0 for all i S. Example. If the network is G = { AB H 2 O, AC H 2 O } then S = {A} and S = {B,C} are the minimal siphons of G.

9 Networks and their Siphons Suppose there are s species that form n complexes. These correspond to monomials c y 1,c y 2,...,c yn in the unknown concentrations c 1,...,c s. Our chemical reaction network is a directed graph G whose nodes are labeled by these monomials. A siphon is a subset S of species such that all reactions in a component of G are shut off when c i = 0 for all i S. Example. If the network is G = { AB H 2 O, AC H 2 O } then S = {A} and S = {B,C} are the minimal siphons of G. Question: Which faces of the positive orthant R s 0 can contain steady states of the reaction network? Proposition. If the face {c R s 0 : c i = 0 for i S} contains a steady state, for some dynamics on G, then S is a siphon.

10 Primary Decomposition Fix the ring R = Q[c 1,...,c s ]/ c 1 c 2 c s. With a chemical reaction network G we associate three ideals M G = c y 1,c y 2,...,c yn J G = c y i c y j : c y i c y j is a reaction of G I G = c y i (c y i c y j ) : c y i c y j is a reaction of G

11 Primary Decomposition Fix the ring R = Q[c 1,...,c s ]/ c 1 c 2 c s. With a chemical reaction network G we associate three ideals M G = c y 1,c y 2,...,c yn J G = c y i c y j : c y i c y j is a reaction of G I G = c y i (c y i c y j ) : c y i c y j is a reaction of G Theorem. The minimal siphons of G are the inclusion-minimal sets {i [s] : c i P} where P runs over the minimal primes of I G. If each connected component of G is strongly connected, then we can replace I G by the simpler ideal J G. If G is strongly connected, then we can replace I G by M G. [Anne Shiu and BSt: Siphons in chemical reaction networks, Bulletin of Mathematical Biology (2010)] Monomial primary decomposition is fast: (Example: 50 species and over 1,000,000 minimal siphons. Macaulay2 computes them in 43 seconds. [B. Roune] )

12 Mass-Action Kinetics The node i of the network G is labeled by the monomial c y i = c y i1 1 cy i2 2 c y is s. Y = (y ij ) is an n s-matrix of non-negative integers. The monomial labels are the entries in the row vector Ψ(c) = ( c y 1, c y 2,..., c yn). Mass-action kinetics specified by the network G and the reaction rates κ ij defines the differential equations dc dt = Ψ(c) A κ Y

13 Mass-Action Kinetics The node i of the network G is labeled by the monomial c y i = c y i1 1 cy i2 2 c y is s. Y = (y ij ) is an n s-matrix of non-negative integers. The monomial labels are the entries in the row vector Ψ(c) = ( c y 1, c y 2,..., c yn). Mass-action kinetics specified by the network G and the reaction rates κ ij defines the differential equations dc dt = Ψ(c) A κ Y, where the n n-matrix A κ is the Laplacian of G. Example. The network G = c y 1 c y 2 c y 3 has the Laplacian κ 12 κ 12 0 A κ = κ 21 κ 21 κ 23 κ 23 0 κ 32 κ 32

14 Steady State Analysis The steady states are the solutions c R s 0 Ψ(c) A κ Y = 0 of the s equations These equations are linear in the parameters κ ij but they are non-linear in the variables c 1,...,c s. Mass action on a network G is a toric dynamical system if there exists a solution c R s >0 of the n equations Ψ(c ) A κ = 0.

15 Steady State Analysis The steady states are the solutions c R s 0 Ψ(c) A κ Y = 0 of the s equations These equations are linear in the parameters κ ij but they are non-linear in the variables c 1,...,c s. Mass action on a network G is a toric dynamical system if there exists a solution c R s >0 of the n equations Ψ(c ) A κ = 0. Theorem (Craciun-Dickenstein-Shiu-St, 2009) For fixed κ, the set of such solutions c is either empty or is a toric variety, i.e. an irreducible variety cut out by binomial equations c u 1 1 cu 2 2 cus s = c v 1 1 cv 2 2 cvs s. For a fixed network G, the set κ for which it is non-empty is also a toric variety, called the moduli space of toric dynamical systems.

16 Toric Varieties connect the linear world and the non-linear world. If we set γ i = log(c i ) then the binomial equation c u 1 1 cu 2 2 cus s = c v 1 1 cv 2 2 cvs s translates into the linear equation (u 1 v 1 )γ 1 + (u 2 v 2 )γ (u s v s )γ s = 0.

17 Toric Varieties connect the linear world and the non-linear world. If we set γ i = log(c i ) then the binomial equation c u 1 1 cu 2 2 cus s = c v 1 1 cv 2 2 cvs s translates into the linear equation (u 1 v 1 )γ 1 + (u 2 v 2 )γ (u s v s )γ s = 0. But binomials have a rich structure that is essential in applications: Test Sets in Integer Programming, Markov Bases in Statistics. In context of bio-chemical reaction networks, the previous theorem elucidates the geometry behind Feinberg s Deficiency Theory, led to recent progress on the Global Attractor Conjecture.

18 Topic 2: CONVEX ALGEBRAIC GEOMETRY A spectrahedron is the intersection of the cone of positive semidefinite matrices with an affine-linear space. Its algebraic representation is a linear combination of symmetric matrices A 0 + x 1 A 1 + x 2 A x m A m 0 ( )

19 Topic 2: CONVEX ALGEBRAIC GEOMETRY A spectrahedron is the intersection of the cone of positive semidefinite matrices with an affine-linear space. Its algebraic representation is a linear combination of symmetric matrices A 0 + x 1 A 1 + x 2 A x m A m 0 ( ) Semidefinite programming is the computational problem of maximizing a linear function over a spectrahedron: Maximize c 1 x 1 + c 2 x c m x m subject to ( ) Example: The smallest eigenvalue of a symmetric matrix A is the solution of the SDP: Maximize x subject to A x Id 0.

20 Multifocal Ellipses The 3-ellipse with foci (0,0),(1,0),(0,1) has the representation 2 3 d + 3x 1 y 1 y 0 y y 1 d + x 1 0 y 0 y 0 0 y 0 d + x + 1 y y 0 0 y y 1 d x y y d + x 1 y 1 y y 0 0 y 1 d x 1 0 y y 0 y 0 d x + 1 y y 0 y y 1 d 3x + 1 This is a convex curve of degree eight. Its interior consists of all points (x,y) where this symmetric 8 8-matrix is positive definite:

21 Ellipses are Spectrahedra Theorem: The polynomial equation defining the m-ellipse has degree 2 m if m is odd and degree 2 m ( m m/2) if m is even. It is the determinant of a symmetric matrix of linear polynomials. This representation extends to m-ellipsoids in arbitrary dimensions. [J. Nie, P. Parrilo, BSt.: Semidefinite representation of the k-ellipse, in Algorithms in Algebraic Geometry, I.M.A. Volumes in Mathematics and its Applications, 146, Springer, New York, 2008, pp ] CONVEX ALGEBRAIC GEOMETRY is the marriage of real algebraic geometry with optimization theory. It concerns convex figures such as ellipses, ellipsoids, and much more... including polyhedra

22 Convex Polyhedra Linear programming is semidefinite programming for diagonal matrices. If A 0,A 1,...,A m are diagonal n n-matrices then A 0 + x 1 A 1 + x 2 A x m A m 0 translates into a system of n linear inequalities in the m unknowns. A spectrahedron defined in this manner is a convex polyhedron:

23 Spectrahedra are Beautiful......and useful too: in rank minimization, compressed sensing etc...

24 Non-Linear Convex Hull Computation Input : { (t,t 2,t 3 ) R 3 : 1 t 1 } y y The convex hull of this curve is a spectrahedron. ( ) ( ) 1 x x y Output : ± 0 x y y z y

25 Global Optimization of Polynomials Let f (x 1,...,x m ) be a polynomial of even degree 2d. We wish to compute the global minimum x of f (x) on R m. This optimization problem is hard. It is equivalent to Maximize λ such that f (x) λ is non-negative on R m. The following relaxtion gives a lower bound: Maximize λ such that f (x) λ is a sum of squares of polynomials. This is much easier. It is a semidefinite program. In practice, the optimal value of the SDP often agrees with the global minimum, and an optimal point x can be recovered. [P.Parrilo, BSt: Minimizing polynomial functions, DIMACS Ser. 60, Amer.Math.Soc., Providence, 2003, pp ]

26 SOS Programming: A Univariate Example Let m = 1, d = 2 and f (x) = 3x 4 + 4x 3 12x 2. Then f (x) λ = ( x 2 x 1 ) 3 2 µ 6 x 2 2 2µ 0 x µ 6 0 λ 1 Our problem is to find (λ,µ) such that the 3 3-matrix is positive semidefinite and λ is maximal.

27 SOS Programming: A Univariate Example Let m = 1, d = 2 and f (x) = 3x 4 + 4x 3 12x 2. Then f (x) λ = ( x 2 x 1 ) 3 2 µ 6 x 2 2 2µ 0 x µ 6 0 λ 1 Our problem is to find (λ,µ) such that the 3 3-matrix is positive semidefinite and λ is maximal. The optimal solution of this SDP is (λ,µ ) = ( 32, 2). Cholesky factorization reveals the SOS representation f (x) λ = ( ( 3 x 4 ) (x + 2) ) 2 8( ) 2. + x We see that the global minimum is x = 2. This approach works for many polynomial optimization problems.

28 Characterization of Spectrahedra A convex hypersurface of degree d in R n is rigid convex if every line passing through its interior meets the corresponding complex hypersurface in d real points. Theorem (Helton Vinnikov (2006)) Every spectrahedron is rigid convex. The converse is true for n = 2. Open problem: Is every compact convex basic semialgebraic set S the projection of a spectrahedron in higher dimensions?

29 Topic 3: TROPICAL MATHEMATICS Let s simplify arithmetic: x y = minimum of x and y x y = x + y Example: 3 (4 5) = = 7 8 = 7

30 Topic 3: TROPICAL MATHEMATICS Let s simplify arithmetic: x y = minimum of x and y x y = x + y Example: 3 (4 5) = = 7 8 = 7 Neutral Elements: x = x 0 x = x

31 Topic 3: TROPICAL MATHEMATICS Let s simplify arithmetic: x y = minimum of x and y x y = x + y Example: 3 (4 5) = = 7 8 = 7 Neutral Elements: x = x 0 x = x Matrix Multiplication: [ ] [ ] = [ ]

32 Matrices and Metrics 1 2 D = 0 d 12 d 13 d 14 d 21 0 d 23 d 24 d 31 d 32 0 d 34 d 41 d 42 d

33 Matrices and Metrics 1 2 D = 0 d 12 d 13 d 14 d 21 0 d 23 d 24 d 31 d 32 0 d 34 d 41 d 42 d 43 0 The (i,j)-entry of the matrix D k = D D D is the length of a shortest path from i to j using k steps. To find shortest pairwise distances in a directed graph with n nodes, compute the tropical matrix power D n. 3 4

34 Matrices and Metrics 1 2 D = 0 d 12 d 13 d 14 d 21 0 d 23 d 24 d 31 d 32 0 d 34 d 41 d 42 d 43 0 The (i,j)-entry of the matrix D k = D D D is the length of a shortest path from i to j using k steps. To find shortest pairwise distances in a directed graph with n nodes, compute the tropical matrix power D n. A symmetric matrix D represents a metric if and only if D 2 = D (triangle inequalities) 3 4

35 Matrices and Metrics 1 2 D = 0 d 12 d 13 d 14 d 21 0 d 23 d 24 d 31 d 32 0 d 34 d 41 d 42 d 43 0 The (i,j)-entry of the matrix D k = D D D is the length of a shortest path from i to j using k steps. To find shortest pairwise distances in a directed graph with n nodes, compute the tropical matrix power D n. A symmetric matrix D represents a metric if and only if D 2 = D (triangle inequalities) D is a tree metric if it comes from a tree with edge lengths d 12 = = 13, etc.

36 Phylogenetics Q: Is every metric a tree metric? A: No, but biologists care about those that are.

37 Phylogenetics Q: Is every metric a tree metric? A: No, but biologists care about those that are. Theorem [4 Point Condition]: A metric D is a tree metric if and only if the max-plus polynomial d ij d kl d ik d jl d il d jk attains its maximum twice, for any four leaves i, j, k and l. Theorem 2.34 in [L.Pachter and BSt: Algebraic Statistics for Computational Biology, Cambridge Univ. Press, 2007] d 12 = 13, d 13 = 11, d 14 = 8, d 23 = 14, d 24 = 9, d 34 = 9. d 12 d 34 d 13 d 24 d 14 d 23 = = 22. Theorem: The space of trees is the tropical Grassmannian G(2, n).

38 Closing Thoughts Linear Algebra has been fantastically useful. Non-Linear Algebra is the natural next step. Quiz: How to draw the convex hull of the trigonometric curve (cos(θ),cos(2θ),sin(3θ) )?

39 Closing Thoughts Linear Algebra has been fantastically useful. Non-Linear Algebra is the natural next step. Quiz: How to draw the convex hull of the trigonometric curve (cos(θ),cos(2θ),sin(3θ) )? Quiz: How many real solutions do these equations have: x 2 + y 2 + z 2 = 2 x 3 + y 3 + z 3 = 3 x 4 + y 4 + z 4 = 4 To find out, please talk to one of the members of the SIAM Activity Group in Algebraic Geometry SI(AG) 2. THANK YOU for your attention!!