Path Signature

Definition

Let $M$ be a smooth manifold with a linearly independent (over $R$ ) collection of 1-forms

A = {a_{1}, . . ., a_{m}} \subset Ω^{1} (M) .

Let $γ \in P (M)$ be a path in $M$ . The path signature $S_{A} (γ)$ of $γ$ with respect to $A$ is an element of the completed tensor algebra $\hat{T} (R [A])$ whose coefficient of each $k$ -word

a_{i_{1}} . . . a_{i_{k}} \in W_{k} (A)

is given by the Chen iterated integral:

\int_{γ} a_{i_{1}} . . . a_{i_{k}} .

Grading

The length of the words indexing the coefficients of $S_{A} (γ)$ induces a grading on the path signature. For each integer $k \geq 1$ the $k$ -th level of the signature is defined by

S_{A}^{k} (γ) = \sum_{a_{i_{1}} . . . a_{i_{k}} \in W_{k} (A)} (\int_{γ} a_{i_{1}} . . . a_{i_{k}}) a_{i_{1}} \otimes . . . \otimes a_{i_{k}} .

By convention, we define $S_{A}^{0} (γ) = 1$ . Therefore the graded expansion of the signature is

S_{A} (γ) = 1 + \sum_{k \geq 1} S_{A}^{k} (γ) .

We will often find it useful to think of the signature as a map $S_{A} : P (M) \to \hat{T} (R [A])$ from the path space of a manifold into the completed tensor algebra defined as above. This allows us to state a number of interesting properties.

Reparameterization Invariance

Given an orientation-preserving diffeomorphism $ϕ : [c, d] \to [a, b]$ and a path $γ : [a, b] \to M,$

S_{A} (γ \circ ϕ) = S_{A} (γ) .

Compatibility with Pullback

Let $M$ and $N$ be smooth manifolds and $F : N \to M$ a diffeomorphism between them. Let

A = {a_{1}, . . ., a_{m}} \subset Ω^{1} (M)

be a linearly independent collection of 1-forms on $M$ . Then the pullback

F^{*} : Ω^{1} (M) \to Ω^{1} (N)

induces a vector space isomorphism

F_{A}^{*} : R [A] \to R [F^{*} (A)]

which extends to an algebra isomorphism

\hat{T} (F_{A}^{*}) : \hat{T} (R [A]) \to \hat{T} (R [F^{*} (A)]) .

Further for every path $γ \in P (N)$ ,

\hat{T} (F_{A}^{*}) (S_{A} (F \circ γ)) = S_{F^{*} (A)} (γ)

or equivalently

S_{A} (F \circ γ) = \hat{T} (F_{A}^{*})^{- 1} (S_{F^{*} (A)} (γ)) .

This can also be viewed as a commutative diagram.

Commutative Diagram (Compatibility with Pullback)

Example

Let $M$ be a smooth manifold with a linearly independent collection of 1-forms $A .$ Then let $F : M \to M$ be a diffeomorphism and $γ \in P (M)$ a path. If for every $a \in A$ $F^{*} (a) = a$ then

S_{A} (F \circ γ) = S_{A} (γ)

A notable example of this is translations $T_{v} : R^{n} \to R^{n}$ when $A$ contains only constant 1-forms.

It is interesting to note here that invariance of the signature is not only a property of the underlying manifold, but also the chosen collection of 1-forms!

Chen's Identity

Let $α, β \in P (M)$ be a pair of composable paths in $M$ with composition $α * β \in P (M)$ . Then

S_{A} (α * β) = S_{A} (α) \otimes S_{A} (β) .

Exposition

Ideas and Working Directions

TeX and Citations

TeX link: (add once this is written up in Dissertation-Writing)