Varieties of Signature Tensors Lecture #1

Transcription

1 Lecture #1 Carlos Enrique Améndola Cerón (TU Munich) Summer School in Algebraic Statistics Tromsø, September 218

2 Setup: Signatures Let X : [, 1] R d be a piecewise differentiable path. Coordinate functions: X 1, X 2,..., X d : R R Their differentials dx i (t) = X i (t)dt are the coordinates of the vector dx = (dx 1, dx 2,..., dx d ) The kth signature of X is a tensor σ (k) (X ) of order k and format d d d. It is the multivariate integral: σ (k) (X ) = dx (t 1 ) dx (t 2 ) dx (t k ), where = { (t 1, t 2,..., t k ) R k : t 1 t 2 t k 1 }. Its d k entries σ i1 i 2...i k are the iterated integrals σ i1 i 2 i k = 1 tk t3 t2 dx i1 (t 1 ) dx i2 (t 2 ) dx ik (t k ).

3 Signatures σ (k) (X ) has entries σ i1 i 2 i k = 1 tk Let s start with k = 1: Fundamental Theorem of Calculus: 1 t3 t2 dx i1 (t 1 ) dx i2 (t 2 ) dx ik (t k ). dx i (t) = X i (1) X i () The first signature of the path X is σ (1) (X ) = 1 dx (t) = X (1) X () R d

4 Signature Matrices Now let s consider k = 2. Then the second signature S = σ (2) (X ) is the d d matrix with entries σ ij = 1 t dx i (s)dx j (t) Set X () =. Applying Fundamental Theorem of Calculus again: We obtain σ ij = 1 X i (t)x j (t)dt σ ij + σ ji = X i (1) X j (1) In matrix notation, S + S T = X (1) X (1) T In particular, the symmetric matrix S + S T has rank one! The skew-symmetric matrix S S T measures deviation from linearity: σ ij σ ji = 1 (X i (t)x j (t) X j (t)x i (t))dt

5 Lévy Area The entry σ ij σ ji of the skew-symmetric matrix S S T is the area below the line minus the area above the line, known as a Lévy area:

6 Some History Introduced by Kuo Tsai Chen in the 195s: K.-T. Chen: Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Annals of Mathematics 65 (1957) K.-T. Chen: Integration of paths a faithful representation of paths by noncommutative formal power series, Transactions AMS 89 (1958) The signature of a path X is the sequence of tensors σ(x ) = ( 1, σ (1) (X ), σ (2) (X ), σ (3) (X ),..., σ (n) (X ),... ) Essential question: how much information does the signature reveal about the path X? Signature determines paths! (modulo starting point, parametrization and tree-like excursion) B. Hambly and T. Lyons: Uniqueness for the signature of a path of bounded variation and the reduced path group, Annals of Mathematics 171 (21)

7 History Signatures are central to the theory of rough paths, a revolutionary view on Stochastic Analysis. P. Friz and N. Victoir: Multidimensional Stochastic Processes as Rough Paths. Theory and Applications, Cambridge University Press, 21. P. Friz and M. Hairer: A Course on Rough Paths. With an introduction to regularity structures, Universitext, Springer, Cham, 214. They can be used to encode and model data! T. Lyons: Rough paths, signatures and the modelling of functions on streams, Proc. International Congress of Mathematicians 214, Seoul I. Chevyrev and A. Kormilitzin: A primer on the signature method in machine learning, arxiv:

8 Present...and Future What happens when signatures meet Algebraic Statistics? C. Améndola, P. Friz and B. Sturmfels: Varieties of Signature Tensors, arxiv: M. Pfeffer, A. Seigal and B. Sturmfels: Learning Paths from Signature Tensors, arxiv: F. Galuppi: The Rough Veronese Variety, arxiv: L. Colmenajero and M. Michalek: Signature Varieties of Axis Paths, in preparation. YOUR papers on signatures! We are interested in projective varieties in tensor space P dk 1 that arise when X ranges over some nice families of paths.

9 x 4 + y 4 + z 4 x 2 y 2 z 2 x 2 y 2 x 2 z 2 y 2 z =

10 Algebraic Varieties Solution set of a polynomial system of equations. V K d (affine) algebraic variety we can find a set of polynomials F K[s 1,..., s d ] such that V = {a K d f (a) = for all f F} If polynomials are homogeneous work in projective space P d 1. The ideal V associated to a variety V: set of all polynomials that vanish on V. Key Fact: a polynomial map σ : K m K d induces naturally an algebraic variety that contains the image of σ.

11 Example of a Signature Variety Let d = 2 and consider quadratic paths in the plane R 2 : X (t) = ( x 11 t + x 12 t 2, x 21 t + x 22 t 2 ) ( ) ( ) T x11 x = 12 t x 21 x 22 t 2 Their kth signature tensors depend polynomially of degree k on x ij. σ (1) (X ) = (σ 1, σ 2 ) = ( x 11 + x 12, x 21 + x 22 ). σ ij = = = 1 t (x i1 + 2x i2 s)ds (x j1 + 2x j2 t)dt 1 (x i1t + x i2 t 2 ) (x j1 + 2x j2 t) dt 1 [x i1x j1 t + (2x i1 x j2 + x i2 x j1 )t 2 + 2x i2 x j2 t 3 ] dt 1 = 2 x i1x j x i1x j x i2x j x i2x j2. We can write σ (2) (X ) as ( ) 1 x11 +x 12 (x11+x ) 1 12, x 21+x (x 11x 22 x 12 x 21 ) x 21 +x 22 ( ) 1. 1

12 Example of a Signature Variety The variety of all such signature matrices is the solution set of the quadratic equation (σ 12 + σ 21 ) 2 4σ 11 σ 22 =. This means that the image variety associated to signature matrices of polynomial paths of degree two in the plane is a hypersurface in P 3. We will denote this surface by P 2,2,2. Its prime ideal generated by the quadric above is P 2,2,2. We want to study and understand these varieties! Question: What is the resulting variety if we restrict to linear paths? Answer: The Veronese variety!

13 The third signature σ (3) (X ) is a tensor (d = 2, k = 3). σ 111 = 1 6 (x 11 + x 12 ) 3 σ 112 = 1 6 (x 11+x 12 ) 2 (x 21 +x 22 ) (5x x 12 )(x 11 x 22 x 12 x 21 ) σ 121 = 1 6 (x 11+x 12 ) 2 (x 21 +x 22 ) (2x 12)(x 11 x 22 x 12 x 21 ) σ 211 = 1 6 (x 11+x 12 ) 2 (x 21 +x 22 ) 1 6 (5x x 12 )(x 11 x 22 x 12 x 21 ) σ 122 = 1 6 (x 11+x 12 )(x 21 +x 22 ) (5x x 22 )(x 11 x 22 x 12 x 21 ) σ 212 = 1 6 (x 11+x 12 )(x 21 +x 22 ) (2x 22)(x 11 x 22 x 12 x 21 ) σ 221 = 1 6 (x 11+x 12 )(x 21 +x 22 ) (5x x 22 )(x 11 x 22 x 12 x 21 ) σ 222 = 1 6 (x 21 + x 22 ) 3 Goal: find the polynomial relations among the eight entries of σ (3) (X ) Image signature variety is P 2,3,2 and its prime ideal is P 2,3,2. An instance of the general P d,k,m of polynomial paths of degree m.

14 Some Favorite Tensors The canonical axis path C axis in R d goes from (,,..., ) to (1, 1,..., 1) in d linear steps in unit directions e 1, e 2,..., e d. Exercise: What is σ (k) (C axis )? Hint: The entry σ i1 i 2...i k is always zero unless i 1 i 2 i k. The canonical monomial path C mono in R d given by t (t, t 2, t 3,..., t d ) goes from (,,..., ) to (1, 1,..., 1) along the rational normal curve. Exercise: What is σ (k) (C mono )? Key observation: We can compute the kth signature tensors of piecewise linear paths or polynomial paths just by knowing σ (k) (C axis ) or σ (k) (C mono ), respectively. How? Encode any such path as a matrix X. The map X σ (k) (X ) is then given by the congruence action X X [[C; X, X,..., X]].

15 The Congruence Action: Example Example (P 2,2,2 revisited) Consider again quadratic paths in R 2 that were given by ) X (t) = ( x 11 t + x 12 t 2, x 21 t + x 22 t 2 ) T = ( x11 x 12 x 21 x 22 The core tensor is the signature matrix σ (2) (C mono ) = ( Then σ (2) (X ) is given by [[C; X, X]] = X σ (2) (C mono ) X T Indeed, we had computed that ( ) ( ) t t t 2 = X t σ ij (X ) = 1 2 x i1x j x i1x j x i2x j x i2x j2. ).

16 The Skyline Path is an axis path with 13 steps in R 2, given by the columns of: [ ] X = Its signature tensor can be obtained by multiplying the core tensor C axis of format with the 2 13 matrix X on all three sides: S skyline = [[C axis ; X, X, X]] = 1 [ ] Three-step path and cubic path with the same signature tensor:

17 Klee-Minty Path X = σ (3) (X ) = [[C axis ; X, X, X]] =