Signatures of paths, the shuffle algebra, and de Bruijns formula - PowerPoint PPT Presentation

Signatures of paths, the shuffle algebra, and de Bruijn’s formula Laura Colmenarejo (UMass Amherst) (Joint work with F. Galuppi & M. Micha� lek, and J. Diehl & M.-S ¸. Sorea) ACPMS – June 19, 2020 L. Colmenarejo (UMass Amherst) ACPMS – June 19, 2020

Motivation → R d A path is a continuous map X : [0 , 1] − ◮ Very simple mathematical object ◮ Tool to interpret a wide range of situations (physical transformations, meteorological models, medical experiments, stock market...) ◮ Downside - being a continuous object, explicit computations on a path are not easy to handle ◮ Solution - find invariants that can provide us enough information ◮ In the 1950s, Chen introduced the iterated-integral signature of a piecewise continuously differentiable path in the research area of stochastic analysis L. Colmenarejo (UMass Amherst) ACPMS – June 19, 2020

Our object: The signature → R d such that A good path is a map X : [0 , 1] − X ( t ) = ( X 1 ( t ) , . . . , X d ( t )) with X i ( t ) a piecewise smooth function for t ∈ [0 , 1] and X i (0) = 0, for all i . For each sequence ( i 1 . . . i k ) ∈ [ d ] k , define (the real number) � 1 � t k � t 3 � t 2 X i 1 ( t 1 ) · . . . · ˙ ˙ X i k ( t k ) dt 1 . . . dt k σ ( i 1 ... i k ) ( X ) := · · · 0 0 0 0 The k -th signature of X is the sequence � σ ( i 1 ... i k ) ( X ) | ( i 1 . . . i k ) ∈ [ d ] k � σ ( k ) ( X ) := , and the signature of X is the sequence σ ( X ) := ( σ ( k ) ( X ) | k ∈ N ) , with σ (0) ( X ) = 1 . L. Colmenarejo (UMass Amherst) ACPMS – June 19, 2020

→ R 2 with X 1 ( t ) = t and X 2 ( t ) = t 2 Path: X : [0 , 1] − � 1 � 1 2 t 1 dt 1 = t 2 � � 1 ˙ X 2 ( t 1 ) dt 1 = σ 2 ( X ) = 0 = 1 0 0 � 1 � t 2 � 1 X 1 ( t 1 ) ˙ ˙ X 2 ( t 2 ) dt 1 dt 2 = t 2 ˙ X 2 ( t 2 ) dt 2 σ 12 ( X ) = 0 0 0 � 1 � 1 2 dt 2 = 2 · t 3 = 2 � 2 t 2 2 = � 3 3 � 0 0 � 1 � t 3 � t 2 X 2 ( t 1 ) ˙ ˙ X 2 ( t 2 ) ˙ X 2 ( t 3 ) dt 1 dt 2 dt 3 σ 222 ( X ) = 0 0 0 � 1 � t 3 2 ˙ X 2 ( t 2 ) ˙ t 2 X 2 ( t 3 ) dt 2 dt 3 = 0 0 � 1 � r 3 � 1 � 1 r 4 3 dr 3 = 1 2 r 3 2 dr 2 dX 2 2 dX 2 3 r 5 = r 3 = r 3 = 6 0 0 0 0 L. Colmenarejo (UMass Amherst) ACPMS – June 19, 2020

Example (continuation) → R 2 with X 1 ( t ) = t and X 2 ( t ) = t 2 Path: X : [0 , 1] − Previous computation: σ 2 ( X ) = 1, σ 12 ( X ) = 2 3, and σ 222 ( X ) = 1 6 The signature of X starts like... σ ( X ) = 1 + 1 · 2 + 1 2 · ( 11 + 22 ) + 1 3 · (2 · 12 + 21 ) + 1 6 · 222 + . . . [Chat comment] Wait, but this is not a sequence... [Chat answer] Yes, you are right. Should we ask? [Speaker’s answer] I want to look at signatures from a more combinatorial perspective and so, I’ll invoque... ✞ ☎ Words ✝ ✆ L. Colmenarejo (UMass Amherst) ACPMS – June 19, 2020

Example (continuation) → R 2 with X 1 ( t ) = t and X 2 ( t ) = t 2 Path: X : [0 , 1] − Previous computation: σ 2 ( X ) = 1, σ 12 ( X ) = 2 3, and σ 222 ( X ) = 1 6 The signature of X starts like... σ ( X ) = 1 + 1 · 2 + 1 2 · ( 11 + 22 ) + 1 3 · (2 · 12 + 21 ) + 1 6 · 222 + . . . [Chat comment] Wait, but this is not a sequence... [Chat answer] Yes, you are right. Should we ask? [Speaker’s answer] I want to look at signatures from a more combinatorial perspective and so, I’ll invoque... ✞ ☎ Words ✝ ✆ L. Colmenarejo (UMass Amherst) ACPMS – June 19, 2020

Example (continuation) → R 2 with X 1 ( t ) = t and X 2 ( t ) = t 2 Path: X : [0 , 1] − Previous computation: σ 2 ( X ) = 1, σ 12 ( X ) = 2 3, and σ 222 ( X ) = 1 6 The signature of X starts like... σ ( X ) = 1 + 1 · 2 + 1 2 · ( 11 + 22 ) + 1 3 · (2 · 12 + 21 ) + 1 6 · 222 + . . . [Chat comment] Wait, but this is not a sequence... [Chat answer] Yes, you are right. Should we ask? [Speaker’s answer] I want to look at signatures from a more combinatorial perspective and so, I’ll invoque... ✞ ☎ Words ✝ ✆ L. Colmenarejo (UMass Amherst) ACPMS – June 19, 2020

Our framework For each sequence ( i 1 . . . i k ) ∈ [ d ] k , define (the real number) � 1 � t k � t 3 � t 2 X i 1 ( t 1 ) · . . . · ˙ ˙ X i k ( t k ) dt 1 . . . dt k . σ ( i 1 ... i k ) ( X ) := · · · 0 0 0 0 Let W k d be the set of words of length k in the alphabet [ d ]. The sequence ( i 1 . . . i k ) ∈ [ d ] k corresponds to a word w ∈ W k d . We encode the signature of X as the sequence � � , with σ (0) = e σ ( X ) := σ w ( X ) · w k ≥ 0 w ∈ W k d � �� � σ ( k ) ( X ) Path: X ( t ) = ( t , t 2 ) 1 + 1 · 2 + 1 2 · ( 11 + 22 ) + 1 3 · (2 · 12 + 21 ) + 1 σ ( X ) = 6 · 222 + . . . 1 + 2 + 1 2 · ( 11 + 22 ) + 1 3 · (2 · 12 + 21 ) + 1 σ ( X ) = 6 · 222 + . . . L. Colmenarejo (UMass Amherst) ACPMS – June 19, 2020

Our framework For each sequence ( i 1 . . . i k ) ∈ [ d ] k , define (the real number) � 1 � t k � t 3 � t 2 X i 1 ( t 1 ) · . . . · ˙ ˙ X i k ( t k ) dt 1 . . . dt k . σ ( i 1 ... i k ) ( X ) := · · · 0 0 0 0 Let W k d be the set of words of length k in the alphabet [ d ]. The sequence ( i 1 . . . i k ) ∈ [ d ] k corresponds to a word w ∈ W k d . We encode the signature of X as the sequence � � , with σ (0) = e σ ( X ) := σ w ( X ) · w k ≥ 0 w ∈ W k d � �� � σ ( k ) ( X ) Path: X ( t ) = ( t , t 2 ) 1 + 1 · 2 + 1 2 · ( 11 + 22 ) + 1 3 · (2 · 12 + 21 ) + 1 σ ( X ) = 6 · 222 + . . . 1 + 2 + 1 2 · ( 11 + 22 ) + 1 3 · (2 · 12 + 21 ) + 1 σ ( X ) = 6 · 222 + . . . L. Colmenarejo (UMass Amherst) ACPMS – June 19, 2020

Our framework ( T (( R d )) , • ): space of formal power series in words in the alphabet { 1 , . . . , d } , together with the concatenation product, w • v ( wv ). ( T ( R d ) , ✁ ): algebra given by the set of polynomials in words in the same alphabet, with the shuffle product, w ✁ v . Example: For w = 12 and 345 , w • v = 12345 w ✁ v = 12345 + 13245 + 13425 + 31245 + 31425 + 34125 + . . . Some comments: ◮ T (( R d )) is a non-commutative algebra ◮ T ( R d ) is a commutative algebra (tensor/shuffle algebra) ◮ Both algebras are graded by the length of the words ◮ The dual pairing in T (( R d )) × T ( R d ): � � w a w · w , v � = a v L. Colmenarejo (UMass Amherst) ACPMS – June 19, 2020

Why these algebras? There are two identities that are very relevant and show that the signature behaves well with respect to these operations. Shuffle identity: For any u , v ∈ T ( R d ), � σ ( X ) , u ✁ v � = � σ ( X ) , u � · � σ ( X ) , v � Chen’s relation: For any two good paths X , Y in R d , define X ⊔ Y to be the concatenation (good) path in R d . Then, σ ( X ⊔ Y ) = σ ( X ) • σ ( Y ) Combinatorial perspective: Hopf algebra L. Colmenarejo (UMass Amherst) ACPMS – June 19, 2020

Why am I interested in singature of paths? ◮ Old topic in Stochastic Analysis ◮ P. K. Fritz (TU Berlin) � couple of textbooks and some more recent work on rough paths . ◮ Data Science and signatures as tensors ◮ M. Pfeffer, B. Sturmfels, & A. Siegal � Given partial information of a signature, can we recover the path? ◮ J. Diehl & J. Reizenstein � Combinatorial approach to understand invariants of multidimensional times series based on signatures (related to representation theory) ◮ C. Am´ endola, B. Sturmfels, & P. K. Fritz � varieties of signatures for piecewise linear paths and for polynomial paths. L. Colmenarejo (UMass Amherst) ACPMS – June 19, 2020

My projects on signatures ◮ Signatures of paths transformed by polynomial maps , with R. Preiß. (Contributions to Algebra and Geometry, 2020) ◮ Related to half-shuffle algebra and the Zinbiel algebras ◮ Toric geometry of path signature varieties , with F. Galuppi and M. Micha� lek (arXiv:1903.03779) ◮ The signature varieties for rough paths ◮ The signature varieties given by the simplest paths we could thing of, axis-parallel paths ◮ Determinant result ◮ A quadratic identity in the shuffle algebra and an alternative proof for de Bruijn’s formula , with J. Diehl and M.-S ¸. Sorea (arXiv: 2003.01574) ◮ Generalization of this determinant result ◮ Alternative proof of de Bruijn’s formulas L. Colmenarejo (UMass Amherst) ACPMS – June 19, 2020

Axis paths A good path X = v 1 ⊔ . . . ⊔ v m is an axis path if there are a 1 , . . . , a m ∈ R such that v i = a i e ν i for every i , where ν i ∈ [ d ]. An axis path is characterized by two sequences ◮ Sequence of lengths: a = ( a 1 , . . . , a m ) ∈ R m , where a i stores the length of the i -th step. ◮ Shape of X : ν = ( ν 1 , . . . , ν m ), where ν i ∈ { 1 , 2 , . . . , d } stores the direction of the i -th step. ◮ Each ν induces a partition π ν = { π 1 | π 2 | . . . | π d } of the set { 1 , . . . , m } , defined by ( π ν ) i = { j ∈ { 1 , . . . , m } | ν j = i } . For ν = (1 , 2 , 1 , 3 , 3 , 1), π ν = { 1 , 3 , 6 | 2 | 4 , 5 } . For ν = (1 , 2 , 1 , 3 , 2 , 3 , 1 , 4), π ν = { 1 , 3 , 7 | 2 , 5 | 4 , 6 | 8 } . L. Colmenarejo (UMass Amherst) ACPMS – June 19, 2020