Varieties of Signature Tensors

Bernd SturmfelsMPI Leipzig and UC Berkeley

Work with Carlos Amendola and Peter Frizand related articles

Paths and their Signatures

A path is a piecewise differentiable map X : [0, 1]→ Rd with X (0) = 0.Coordinates X1, . . . ,Xd : R→ R. Differentials dXi (t) = X ′i (t)dt.

The kth signature of X is a tensor σ(k)(X ) of format d× · · ·×d .

Its dk entries σi1i2···ik are multivariate integrals over the simplex

∆ ={

(t1, t2, . . . , tk) ∈ Rk : 0 ≤ t1 ≤ t2 ≤ · · · ≤ tk ≤ 1}.

Definition [iterated integrals]:

σi1i2···ik =

∫ 1

0

∫ tk

0

· · ·∫ t3

0

∫ t2

0

dXi1 (t1)dXi2 (t2) · · · dXik−1(tk−1)dXik (tk).

Fundamental Theorem of Calculus: σ(1)(X ) = X (1) ∈ Rd

We study tensors that arise when X ranges over nice families of paths.

Paths and their Signatures

A path is a piecewise differentiable map X : [0, 1]→ Rd with X (0) = 0.Coordinates X1, . . . ,Xd : R→ R. Differentials dXi (t) = X ′i (t)dt.

The kth signature of X is a tensor σ(k)(X ) of format d× · · ·×d .

Its dk entries σi1i2···ik are multivariate integrals over the simplex

∆ ={

(t1, t2, . . . , tk) ∈ Rk : 0 ≤ t1 ≤ t2 ≤ · · · ≤ tk ≤ 1}.

Definition [iterated integrals]:

σi1i2···ik =

∫ 1

0

∫ tk

0

· · ·∫ t3

0

∫ t2

0

dXi1 (t1)dXi2 (t2) · · · dXik−1(tk−1)dXik (tk).

Fundamental Theorem of Calculus: σ(1)(X ) = X (1) ∈ Rd

We study tensors that arise when X ranges over nice families of paths.

Planar Example

Consider quadratic paths in the plane R2:

X (t) = (x11t + x12t2, x21t + x22t

2)T =

(x11 x12

x21 x22

)·(tt2

)

The third signature σ(3)(X ) is a 2×2×2 tensor. Its entries are

σ111 = 16 (x11 + x12)3

σ112 = 16 (x11+x12)2(x21+x22) + 1

60 (5x11 + 4x12)(x11x22 − x21x12)

σ121 = 16 (x11+x12)2(x21+x22) + 1

60 (2x12)(x11x22 − x21x12)

σ211 = 16 (x11+x12)2(x21+x22) − 1

60 (5x11 + 6x12)(x11x22 − x21x12)

σ122 = 16 (x11+x12)(x21+x22)2 + 1

60 (5x21 + 6x22)(x11x22 − x21x12)

σ212 = 16 (x11+x12)(x21+x22)2 − 1

60 (2x22)(x11x22 − x21x12)

σ221 = 16 (x11+x12)(x21+x22)2 − 1

60 (5x21 + 4x22)(x11x22 − x21x12)

σ222 = 16 (x21 + x22)3

This parametrizes a threefold of degree six in P7 = P(C2 ⊗ C2 ⊗ C2).

Signature MatricesThe d × d matrix S = σ(2)(X ) has entries

σij =

∫ 1

0

∫ t

0dXi (s) dXj(t) =

∫ 1

0Xi (t)X ′j (t)dt.

The symmetric matrix S + ST has rank one: σij + σji = Xi (1) · Xj(1).

Levy Area

Figure 5: Example of signed Levy area of a curve. Areas above and under the chord

connecting two endpoints are negative and positive respectively.

1.3 Important properties of signature

We now review several fundamental properties of the signature of paths. We do not provide

all the details behind the proofs of these properties, but we emphasize that they are all

straightforward consequences of classical integration theory. Several deeper results are

discussed in the following Section 1.4, but only on an informal level.

1.3.1 Invariance under time reparametrisations

We call a surjective, continuous, non-decreasing function : [a, b] 7! [a, b] a reparametriza-

tion. For simplicity, we shall only consider smooth reparametrizations, although, just like

in the definition of the path integral, this is not strictly necessary.

Let X, Y : [a, b] 7! R be two real-valued paths and : [a, b] 7! [a, b] a reparametrization.

Define the paths eX, eY : [a, b] 7! R by eXt = X (t) and eYt = Y (t). Observe that

eXt = X (t) (t), (1.36)

from which it follows that

Z b

a

eYtd eXt =

Z b

aY (t)X (t) (t)dt =

Z b

aYudXu, (1.37)

where the last equality follows by making the substitution u = (t). This shows that path

integrals are invariant under a time reparametrization of both paths.

– 11 –

The skew-symmetric matrix S − ST measures deviation from linearity:

σij − σji =

∫ 1

0

(Xi (t)X ′j (t)− Xj(t)X ′i (t)

)dt

Varieties of Signature MatricesSquare matrix S = symmetric matrix P plus skew-symmetric matrix Q:

S = P +Q, where P =1

2(S +ST ) and Q =

1

2(S−ST ).

TheoremFor each d and m, the following subvarieties of Pd2−1 coincide:

1. Signature matrices of piecewise linear paths with m segments.

2. Signature matrices of polynomial paths of degree m.

3. Matrices S satisfying

rank(P) ≤ 1 and rank([P Q ]

)≤ m.

These varieties form nested families:

Md ,1 ⊂Md ,2 ⊂Md ,3 ⊂ · · · ⊂ Md ,d =Md ,d+1 = · · · = Ud ,2

Md,m is irreducible of dimension md −(m2

)− 1. Singular locus Md,m−1.

My Favorite Tensors

Example (Canonical Axis Path)

Let Caxis be the path from (0, 0, . . . , 0) to (1, 1, . . . , 1) with dlinear steps in basis directions e1, e2, . . . , ed . The entry σi1i2···ikof σ(k)(Caxis) is zero unless i1 ≤ i2 ≤ · · · ≤ ik .

In that case, it equals 1/k! times the number of distinctpermutations of the string i1i2 · · · ik . Examples for k = 4:

σ1111 = 124 , σ1112 = 1

6 , σ1122 = 14 , σ1123 = 1

2 , σ1234 = 1, σ1243 = 0.

Example (Canonical Monomial Path)

Let Cmono be the path t 7→ (t, t2, t3, . . . , td). It travels from(0, 0, . . . , 0) to (1, 1, . . . , 1) along the rational normal curve.Entries of σ(k)(Cmono) are

σi1i2···ik =i1i1· i2i1 + i2

· i3i1 + i2 + i3

· · · iki1 + i2 + · · ·+ ik

.

The Skyline Pathis an axis path with 13 steps in R2:

X =

[1 0 1 0 1 0 1 0 1 0 1 0 10 1 0 −1 0 2 0 −2 0 1 0 −1 0

]

Equivariance: Obtain 2×2×2 signature tensor by multiplying the13×13×13 tensor Caxis with the 2×13 matrix X on all three sides:

Sskyline = [[Caxis;X ,X ,X ]] =1

6

[343 0 −84 1884 18 −36 0

].

Three-step path and cubic path with the same signature:

Klee-Minty PathX =

1 0 −1 0 1 0 −10 1 0 0 0 −1 00 0 0 1 0 0 0

.

σ(3)(X ) =1

6

0 0 0 0 0 0 0 6 00 0 0 0 0 −6 −6 3 30 6 0 −6 3 −3 0 0 1

ContextP. Friz and M. Hairer: A Course on Rough Paths. With an introductionto regularity structures, Universitext, Springer, Cham, 2014.

P. Friz and N. Victoir: Multidimensional Stochastic Processes as RoughPaths. Theory and Applications, Cambridge University Press, 2010.

T. Lyons: Rough paths, signatures and the modelling of functions onstreams, Proc. International Congress of Mathematicians 2014, Seoul

I. Chevyrev and A. Kormilitzin: A primer on the signature method inmachine learning (2016)

K.-T. Chen: Integration of paths, geometric invariants and a generalizedBaker-Hausdorff formula, Annals of Mathematics 65 (1957)

R. Hain: Iterated integrals and algebraic cycles (2000)

M. Kapranov: Noncommutative geometry and path integrals (2009)

J. Balakrishnan: Iterated Coleman integration for hyperelliptic curves (2013)

L. Colmenarejo and R. Preiss: Signatures of paths transformed bypolynomial maps (2019)

Universal VarietyThe kth signature tensor of a path X in Rd is a point σ(k)(X ) in Pdk−1.

As X ranges over all paths [0, 1]→ Rd ,

these tensors parametrize the universal variety Ud ,k ⊂ Pdk−1.

TheoremThe dimension of the universal variety Ud ,k is the number ofLyndon words of length ≤ k over the alphabet {1, 2, . . . , d}.

A word is a Lyndon word if it is strictly smallerin lexicographic order than all of its rotations.

d \ k 2 3 4 5 6 7 8 92 2 4 7 13 22 40 70 1263 5 13 31 79 195 507 1317 35014 9 29 89 293 963 3303 11463 40583

Table: The dimension of Ud,k is much smaller than dk − 1.

TensorsThe truncated tensor algebra is a non-commutative algebra:

T n(Rd) =n⊕

k=0

(Rd)⊗k

This is also a Lie algebra via

[P,Q] = P ⊗ Q − Q ⊗ P for P,Q ∈ T n(Rd).

Words of length ≤ n on {1, 2, . . . , d} give standard basis

ei1i2···ik := ei1 ⊗ ei2 ⊗ · · · ⊗ eik and its dual σi1i2···ik

T n(Rd)∗ is a commutative algebra with shuffle product tt.

The shuffle product of two words of lengths r and sis the sum over all

(r+ss

)interleavings the two words:

σ12 ttσ34 = σ12tt 34 = σ1234 + σ1324 + σ1342 + σ3124 + σ3142 + σ3412

σ3tt134 = σ3134 + 2σ1334 + σ1343 σ21tt 21 = 2σ2121 + 4σ2211

Free Lie AlgebraLien(Rd) is the smallest Lie subalgebra of T n(Rd) containing Rd .

LemmaThis is characterized by the vanishing of all shuffle linear forms:

Lien(Rd) ={P ∈ T n

0 (Rd) : σI tt J(P) = 0 for all words I , J}.

TheoremThe Lie bracketings of all Lyndon words are a basis for Lien(Rd).

[C. Reutenauer: Free Lie Algebras, Oxford University Press, 1993]

Example. Lie4(R2) is 8-dimensional in T 40 (R2) ' R30. The eight

Lyndon words 1, 2, 12, 112, 122, 1112, 1122, 1222 determine a basis:

[e1, e2] = e12 − e21, . . . , [[[e1, e2], e2], e2] = e1222 − 3e2122 + 3e2212 − e2221

The 22-dim’l space of linear relations is spanned by shuffles

σ21tt 21 = 2σ2121 + 4σ2211 , σ1tt 111 = 4σ1111,σ12tt 21 = 2σ1221 + σ1212 + σ2121 + 2σ2112.

Free Lie GroupTwo polynomial maps on T n

0 (Rd):

exp(P) =∑

r≥0

1

r !P⊗r and log(1 + P) =

∑

r≥1

(−1)r−1

rP⊗r .

The logarithm inverts the exponential function:

log(exp(P)) = P for all P ∈ T n0 (Rd).

The free Lie group is the image of the free Lie algebra:

Gn(Rd) := exp(Lien(Rd)).

TheoremThis Lie group is an algebraic variety in T n

1 (Rd). Its is defined by

σI tt J(P) = σI (P)σJ(P) for all words I , J with |I |+ |J| ≤ n.

Our contribution: This is the prime ideal. It has a nice Grobner basis.

Back to PathsConnection to paths comes from

Theorem (Chen-Chow)

The Lie group Gn(Rd) is precisely the imageof the signature map applied to all paths in Rd :

Gn(Rd) ={σ≤n(X ) : X : [0, 1]→ Rd any smooth path

}

Key Lemma: If X is the piecewise linear path with stepsX1,X2, . . . ,Xm in Rd then the n-step signature of X equals

σ≤n(X ) = exp(X1) ⊗ exp(X2) ⊗ · · · ⊗ exp(Xm) ∈ T n(Rd).

Example (k = 3)The third signature is the d×d×d tensor σ(3)(X ) =

1

6·

m∑

i=1

X⊗3i +

1

2·∑

1≤i<j≤m

(X⊗2i ⊗Xj + Xi ⊗X⊗2

j

)+

∑

1≤i<j<l≤m

Xi ⊗Xj ⊗Xk .

Universal VarietyWe focus on tensors of a fixed order k.

The projection of the Lie group Gd ,k into (Rd)⊗k is a cone.

The resulting variety in Pdk−1 is the universal variety Ud ,k .

Example: Ud,2 ={d×d matrices whose symmetric part has rank 1

}.

With Amendola and Friz, we conjectured that Ud,k is defined by quadrics.

d k amb dim degree quadrics

2 3 7 4 4 6

2 4 15 7 12 33

2 5 31 13 40 150

3 3 26 13 24 81

3 4 80 31 672 954

Francesco Galuppi found a change of coordinates which turnsUd ,k into a toric variety. With this, he disproved our conjecture.

[F. Galuppi: The Rough Veronese Variety, September 2018]

Polynomial Signature Varieties

Consider paths X : [0, 1]→ Rd whose coordinates are polynomials

Xi (t) = xi1t + xi2t2 + xi3t

3 + · · ·+ ximtm.

Identify paths with d ×m-matrices X = (xij).

The kth signature σ(k)(X ) is a d× · · ·×d tensor.It is computed by multiplying the m× · · ·×m tensor

σ(k)(Cmono) =[i1i1· i2i1+i2· i3i1+i2+i3

· · · iki1+i2+···+ik

]

on all k sides with the d ×m matrix X .

The polynomial signature variety Pd ,k,m is the image of

σ(k) : Pdm−1 99K Pdk−1 , X 7→ σ(k)(X ).

Remark: If m ≤ d then this is the closure of a GL(d) orbit in (Cd)⊗m.

Example: Quadratic Paths in 3-Space

The third signature variety P3,3,2 for quadratic paths in R3 hasdimension 5, degree 90, and is cut out by 162 quadrics in P25.

U3,3 has dimension 13, degree 24, and 81 quadrics.

The linear span of P3,3,2 is the hyperplane

σ123 − σ132 − σ213 + σ231 + σ312 − σ321 = 0.

This linear form is the signed volume of the convex hull of a path.

Piecewise Linear Signature Varieties

Piecewise linear paths are also represented by d ×m matrices X .

The kth signature σ(k)(X ) is a d× · · ·×d tensor. It can becomputed by multiplying the upper triangular m× · · ·×mtensor σk(Caxis) on all k sides with the d ×m matrix X .

The piecewise linear signature variety Ld ,k,m is the image of

σ(k) : Pdm−1 99K Pdk−1 , X 7→ σ(k)(X ).

Remark: If m ≤ d then this is the closure of a GL(d) orbit in (Cd)⊗m.

Inclusions and Differences

TheoremFor any d and k , we have the following chains of inclusionsbetween the kth Veronese variety and the kth universal variety:

νk(Pd−1) = Ld,k,1 ⊂ Ld,k,2 ⊂ · · · ⊂ Ld,k,M−1 ⊂ Ld,k,M = Ud,k ⊂ Pdk−1

νk(Pd−1) = Pd,k,1 ⊂ Pd,k,2 ⊂ · · · ⊂ Pd,k,M−1 ⊂ Pd,k,M = Ud,k ⊂ Pdk−1

TheoremFor k ≥ 3 we have L2,k,2 6= P2,k,2 in P2k−1.

Example (k = 4)Use invariant theory to distinguish these threefolds in P15.SL(2)-invariant linear forms on (C2)⊗4 are spanned by

`1 = σ1212 − σ1221 − σ2112 + σ2121

and `2 = σ1122 − σ1221 − σ2112 + σ2211.

`1/`2 is a rational function on P15 that is constant on orbits.It has value 0 on Caxis and value 1/5 on Cmono.

Identifiability λd,k = # Lyndon words

Parameter count shows

dim(Ld ,k,m) ≤ min{λd ,k − 1, dm − 1},and dim(Pd ,k,m) ≤ min{λd ,k − 1, dm − 1}.

If dim = dm − 1 then the variety is algebraically identifiable: themap from d×m matrices X to signature tensors σ(k)(X ) is r -to-1.

If r = 1 then it is birational, and the variety is rationally identifiable.

Conjecture

Let k ≥ 3 and m strictly less than the threshold M =⌊λd,k

d

⌋.

Both varieties Pd ,k,m and Ld ,k,m are rationally identifiable.

Theorem

I Rational identifiability holds for m ≤ 7

I Algebraic identifiability holds for m ≤ 30.

I Identifiability holds for Ld ,k,m provided m ≤ d .

BorderlineIdentifiability is delicate for λd,k = md , when the signature variety

exactly fills the universal variety. We conjecture algebraic identifiability.

Example (d = 2, k = 4,M = 4)

The 7-dim’l variety P2,4,4 = L2,4,4 = U2,4 has degree 12 in P15.Have two parametrizations from P7 of 2×4 matrices. Map fromquartic paths is 48-to-1. From four-segment paths it is 4-to-1.

Consider the four-segment path in R2 given by

X =

[29 15 13 223 26 6 27

]

Three other paths have the same 2× 2× 2× 2 signature tensor:

[36.74838 −17.80169 37.75532 2.2979927.39596 −9.82926 40.23084 24.20246

],

[102.16286 −131.13298 85.92484 2.04528104.55786 −136.84738 86.56467 27.72484

],

[38.53237 38.8057 −79.20533 60.8673528.69523 82.7734 −147.7839 118.3152

].

Computing Shortest Paths

[ M. Pfeffer, A. Seigal, B.St: Learning Paths from Signature Tensors ]

Next summer in Bern

SIAM AG 19 Minisymposia

158 Sessions, two of them on

Algebraic methods in stochastic analysisOrg: Carlos Amendola, Joscha Diehl, Francesco Galuppi and Anna Seigal