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

A path is a continuous map X : [0, 1] − → Rd

◮ 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

• n 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

• f a piecewise continuously differentiable path in the research

area of stochastic analysis

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

Our object: The signature

A good path is a map X : [0, 1] − → Rd such that 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 (i1 . . . ik) ∈ [d]k, define (the real number) σ(i1...ik)(X) := 1 tk · · · t3 t2 ˙ X i1(t1) · . . . · ˙ X ik(tk)dt1 . . . dtk The k-th signature of X is the sequence σ(k)(X) :=

• σ(i1...ik)(X) | (i1 . . . ik) ∈ [d]k

, 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

Path: X : [0, 1] − → R2 with X 1(t) = t and X 2(t) = t2 σ2(X) = 1 ˙ X 2(t1)dt1 = 1 2t1dt1 = t2 1

0 = 1

σ12(X) = 1 t2 ˙ X 1(t1) ˙ X 2(t2)dt1dt2 = 1 t2 ˙ X 2(t2)dt2 = 1 2t2

2dt2 = 2 · t3 2

3

• 1

= 2 3 σ222(X) = 1 t3 t2 ˙ X 2(t1) ˙ X 2(t2) ˙ X 2(t3)dt1dt2dt3 = 1 t3 t2

2 ˙

X 2(t2) ˙ X 2(t3)dt2dt3 = 1 r3 2r3

2 dr2dX 2 r3 =

1 r4

3

2 dX 2

r3 =

1 r5

3 dr3 = 1

6

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

Example (continuation)

Path: X : [0, 1] − → R2 with X 1(t) = t and X 2(t) = t2 Previous computation: σ2(X) = 1, σ12(X) = 2 3, and σ222(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)

Path: X : [0, 1] − → R2 with X 1(t) = t and X 2(t) = t2 Previous computation: σ2(X) = 1, σ12(X) = 2 3, and σ222(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)

Path: X : [0, 1] − → R2 with X 1(t) = t and X 2(t) = t2 Previous computation: σ2(X) = 1, σ12(X) = 2 3, and σ222(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 (i1 . . . ik) ∈ [d]k, define (the real number) σ(i1...ik)(X) := 1 tk · · · t3 t2 ˙ X i1(t1) · . . . · ˙ X ik(tk)dt1 . . . dtk. Let Wk

d be the set of words of length k in the alphabet [d].

The sequence (i1 . . . ik) ∈ [d]k corresponds to a word w ∈ Wk

d.

We encode the signature of X as the sequence σ(X) :=

• k≥0
• w∈Wk

d

σw(X) · w

• σ(k)(X)

, with σ(0) = e Path: X(t) = (t, t2) σ(X) = 1 + 1 · 2 + 1 2 · (11 + 22) + 1 3 · (2 · 12 + 21) + 1 6 · 222 + . . . σ(X) = 1 + 2 + 1 2 · (11 + 22) + 1 3 · (2 · 12 + 21) + 1 6 · 222 + . . .

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

Our framework

For each sequence (i1 . . . ik) ∈ [d]k, define (the real number) σ(i1...ik)(X) := 1 tk · · · t3 t2 ˙ X i1(t1) · . . . · ˙ X ik(tk)dt1 . . . dtk. Let Wk

d be the set of words of length k in the alphabet [d].

The sequence (i1 . . . ik) ∈ [d]k corresponds to a word w ∈ Wk

d.

We encode the signature of X as the sequence σ(X) :=

• k≥0
• w∈Wk

d

σw(X) · w

• σ(k)(X)

, with σ(0) = e Path: X(t) = (t, t2) σ(X) = 1 + 1 · 2 + 1 2 · (11 + 22) + 1 3 · (2 · 12 + 21) + 1 6 · 222 + . . . σ(X) = 1 + 2 + 1 2 · (11 + 22) + 1 3 · (2 · 12 + 21) + 1 6 · 222 + . . .

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

Our framework

(T((Rd)), •): space of formal power series in words in the alphabet {1, . . . , d}, together with the concatenation product, w • v (wv). (T(Rd), ✁): 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((Rd)) is a non-commutative algebra ◮ T(Rd) is a commutative algebra (tensor/shuffle algebra) ◮ Both algebras are graded by the length of the words ◮ The dual pairing in T((Rd)) × T(Rd): w aw · w, v = av

• 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(Rd), σ(X), u ✁ v = σ(X), u · σ(X), v Chen’s relation: For any two good paths X, Y in Rd, define X ⊔ Y to be the concatenation (good) path in Rd. 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

• n 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 = v1 ⊔ . . . ⊔ vm is an axis path if there are a1, . . . , am ∈ R such that vi = aieνi for every i, where νi ∈ [d]. An axis path is characterized by two sequences

◮ Sequence of lengths: a = (a1, . . . , am) ∈ Rm, where ai stores

the length of the i-th step.

◮ Shape of X: ν = (ν1, . . . , νm), where νi ∈ {1, 2, . . . , d} stores

the direction of the i-th step.

◮ Each ν induces a partition πν = {π1|π2| . . . |πd} of the set

{1, . . . , m}, defined by (πν)i = {j ∈ {1, . . . , m} | νj = i}. For ν = (1, 2, 1, 3, 3, 1), πν = {1, 3, 6 | 2 | 4, 5}. For ν = (1, 2, 1, 3, 2, 3, 1, 4), πν = {1, 3, 7|2, 5|4, 6|8}.

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

The signature of axis paths

Theorem [C.–Galuppi–Micha lek] Given an axis path X = (a, ν) = (a, π), σi1...ik(X) =

• (j1,...,jk)

1 s1!s2! · · · sm!aj1aj2 · · · ajk, summing over all the non-decreasing sequences such that jl ∈ πil for l ∈ [k], and sl = #{l = ji|i ∈ [k]}. Example: X a = (a1, . . . , a8), ν = (1, 2, 1, 3, 2, 3, 1, 4), for which πν = {1, 3, 7|2, 5|4, 6|8} σ1234 = a1a2a4a8 + a1a2a6a8 + a1a5a6a8 + a3a5a6a8, σ2314 = a2a4a7a8 + a2a6a7a8 + a5a6a7a8, σ4123 = 0, σ1124 = 1 2a2

1a2a8 + 1

2a2

1a5a8 + a1a3a5a8 + 1

2a2

3a5a8.

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

Determinants of signatures

For k = 2, we can write the signature as a square d × d matrix σ(2)(X) =      σ11 σ12 . . . σ1d σ21 σ22 . . . σ2d . . . . . . ... . . . σd1 σd2 . . . σdd      Question: Would the determinant give us some extra useful information? Answer: Maybe? Let’s do some experiments.

✓ ✒ ✏ ✑

as the square of a polynomial in the ai’s we could always describe det(σ(2)(X)) Experiments showed that up to some scalar factor,

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

Theorem [C.–Galuppi–Micha lek] (the determinant result) Given an axis path X = (ν, a) in Rd, 2d det

• σ(2)(X)
• = P(a)2,

with P(a) :=

• µ=(νi1,...,νid )

(sgnµ)

d

• j=1

aij, where the sum is taken over all subsequences µ = (νi1, . . . , νid) such that i1 < · · · < id and {νi1, . . . , νid} = {1, . . . , d}, and sgnµ is defined as the sign of the permutation (νi1, . . . , νid) ∈ Sd. Some comments:

◮ Technical proof based on the combinatorial description of the

signature of axis paths.

◮ There is a way to understand this determinant in terms of the

shuffle identity.

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

Path X: a = (a1, . . . , a6), ν = (1, 2, 1, 3, 2, 3), π = {1, 3|2, 5|4, 6} We want to compute the determinant of the matrix σ(2)(X)  

1 2a2 1 + a1a3 + 1 2a2 3

a1a2 + a1a5 + a3a5 a1a4 + a3a4 + a1a6 + a3a6 a2a3

1 2a2 2 + a2a5 + 1 2a2 5

a2a4 + a2a6 + a5a6 a4a5

1 2a2 4 + a4a6 + 1 2a2 6

  To compute the monomials in P(a), we have this table {i1, i2, i3} µ sgnµ (i1, i2, i3) µ sgnµ {1, 2, 4} (1, 2, 3) + {1, 2, 6} (1, 2, 3) + {1, 4, 5} (1, 3, 2) − {1, 5, 6} (1, 2, 3) + {2, 3, 4} (2, 1, 3) − {2, 3, 6} (2, 1, 3) − {3, 4, 5} (1, 3, 2) − {3, 5, 6} (1, 2, 3) + det σ(2)(X) = 1 23 (a1a2a4 + a1a2a6 − a1a4a5 + a1a5a6 − a2a3a4 − a2a3a6 − a3a4a5 + a3a5a6)

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

Let X be any good path in R2. Then, det(σ(2)(X)) = det σ11(X) σ12(X) σ21(X) σ22(X)

• = σ11(X) · σ22(X) − σ12(X) · σ21(X)

= σ(X), 11 · σ(X), 22 − σ(X), 12 · σ(X), 21 = σ(X), 11 ✁ 22 − 12 ✁ 21 = 1 4σ(X), (12 − 21)✁2. Define invd :=

• ρ∈Sd

sgn(ρ)ρ(1) . . . ρ(d) ∈ T(Rd). Theorem [C.–Galuppi–Micha lek] (the determinant result ii) For an axis path X = (ν, a) in Rd, P(a) = σ(X), invd, and 2d det

• σ(2)(X)
• = (σ(X), invd)2
• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

More general version

Consider the following matrix in the shuffle algebra Wd =      11 12 . . . 1d 21 22 . . . 2d . . . . . . ... . . . d1 d2 . . . dd      Let det✁(A) be its determinant, where the product is the shuffle. Then, 2d det✁(A) = (invd)✁2. But wait... There are no paths or signatures here!

✞ ✝ ☎ ✆

Could we prove this result without using paths? Yes, we can. To do so, we need to understand invd from other perspective.

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

invd as an invariant

Weyl studied the basis of SL(Rd)-invariants in T(Rd).

◮ Basis indexed by standard Young tableaux (SYT) of shape

(wd), for w ≥ 1. For us, w = 1, 2. Given such a SYT T, let n = wd and consider the word w = j1j2 . . . jn, where jℓ = i if and only if ℓ is in the ith row of T. For instance, T = 1 2 3 4 gives the word 1122.

◮ The basis is {inv(T)}T, with inv(T) := ρ sign(ρ)ρ(w),

where the sum is over all permutations ρ ∈ Sn that leave the values in each column of T unchanged. For instance, inv

• 1

2 3 4

• = (id −(13) − (24) + (13)(24))1122 =

1122 − 2112 − 1221 + 2211.

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

Theorem [C.–Diehl–Sorea] (main result) For d ≥ 1 the following two equalities hold: det✁    11 . . . 1d . . . ... . . . d1 . . . dd    = inv       1 2 3 4 . . . . . .

2d − 1 2d

      = 1 2d inv       1 2 . . . d      

✁2

Example: For d = 2, we have that det✁ 11 12 21 22

• = 1122 + 2211 − 1221 − 2112

= inv   1 2 3 4   = 1 4 inv   1 2  

✁2

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

Proof of First equality

Theorem [C.–Diehl–Sorea] det✁(Wd) =

• σ∈Sd

sgn(σ)✁

d i=1iσ(i)

=

• σ∈Sd

sgn(σ)

τ∈Sd d

• i=1

τ(i)σ(τ

• i)
• = inv

      1 2 3 4 . . . . . .

2d − 1 2d

      Idea of the proof:

◮ Explain the cancellations for the case d = 3 with half-shuffle

and concatenation products for words of length 2

◮ Generalize your argument for any d

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

Proof of Second equality

inv       1 2 3 4 . . . . . .

2d − 1 2d

      = 1 2d inv       1 2 . . . d      

✁2

Idea of the proof:

• 1. Show that the two terms lie in the same 1-dimensional

isotypic component of a representation of a certain subgroup

• f S2d, and that this isotypic component has dimension 1.

◮ The subgroup of S2d is

H :=

• (1, 3), (3, 5), . . . , (2d − 3, 2d − 1), (2, 4), (4, 6), . . . , (2d − 2, 2d)
• .

◮ The isotypic component related of the sign-representation

when we restrict to the subgroup.

• 2. Determine the pre-factors, by looking at a particular word.
• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

Pfaffian of a matrix

Let A = (aij) ∈ Ad×d be a skew-symmetric matrix, where d is even and A is a commutative algebra. The Pfaffian of A is defined by PfA[A] := 1 2d/2 1 (d/2)!

• π∈Sd

sign(π)

d/2

• i=1

aπ(2i−1)aπ(2i) Property: detA[A] = (PfA[A])2 Recall: Any matrix M can be written as M = Sym[M] + Anti[M], with Sym[M] := 1 2(M + M⊤), Anti[M] := 1 2(M − M⊤).

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

de Bruijn’s formula (even case)

Theorem [de Bruijn & C.–Diehl–Sorea] inv       1 2 . . . d       = 2d/2Pf✁[Anti[Wd]]. Example: For d = 4, our result together with de Bruijn’s formula give that det✁

    11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44     = det✁     12 − 21 13 − 31 14 − 41 21 − 12 23 − 32 24 − 42 31 − 13 32 − 23 34 − 43 41 − 14 42 − 24 43 − 34    

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

de Bruijn’s formula (odd case)

Theorem [de Bruijn & C.–Diehl–Sorea] inv       1 2 . . . d       = Pf✁[Zd], for Zd =        2 Anti[Wd] 1 2 . . . d −1 −2 · · · −d        Example: For d = 2, our result together with de Bruijn’s formula give that det✁   11 12 13 21 22 23 31 32 33   = det✁     12 − 21 13 − 31 1 21 − 12 23 − 32 2 31 − 13 32 − 23 3 −1 −2 −3    

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020

Thank you very much!

• L. Colmenarejo (UMass Amherst)

ACPMS – June 19, 2020