# General Signature Kernels

Thomas Cass  ,  Terry Lyons  and  Xingcheng Xu

###### Abstract.

Suppose that $\gamma$ and $\sigma$ are two continuous bounded variation paths
which take values in a finite-dimensional inner product space $V$. The recent
papers [18] and [6] respectively introduced the truncated and the untruncated
signature kernel of $\gamma$ and $\sigma$ and showed how these concepts can be
used in classification and prediction tasks involving multivariate time
series. In this paper we consider general signature kernels of the form

(0.1) $K^{\gamma,\sigma}_{\phi}\left(s,t\right)=\left\langle
S\left(\gamma\right)_{a,s},S\left(\sigma\right)_{a,t}\right\rangle_{\phi}:=\sum_{k=0}^{\infty}\phi(k)\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}$

where $\left\langle\cdot,\cdot\right\rangle_{k}$ is the Hilbert-Schmidt inner-
product on $V^{\otimes k}$ and $\phi:\mathbb{N}\cup\\{0\\}\mapsto\mathbb{C}$.
We show how $K^{\gamma,\sigma}_{\phi}$ can be interpreted in many examples as
an average of PDE solutions and thus how it can estimated computationally
using suitable quadrature formulae. We extend this analysis to derive closed-
form formulae for expressions involving the expected (Stratonovich) signature
of Brownian motion. In doing so we articulate a novel connection between
signature kernels and the hyperbolic development, the latter of which has been
a broadly useful tool in the analysis of the signature, see e.g. [16], [26]
and [2]. As an application we evaluate the use different general signature
kernels as the basis for non-parametric goodness-of-fit tests to Wiener
measure on path space.

Imperial College London and the Alan Turing Institute,
<EMAIL_ADDRESS>
University of Oxford and the Alan Turing Institute<EMAIL_ADDRESS>
Imperial College London<EMAIL_ADDRESS>
The work of all three authors was supported by the EPSRC Programme Grant
EP/S026347/1

Keywords: The signature, expected signatures, kernel methods, general
signature kernels,

Gaussian quadrature, hyperbolic development, contour integration

##  1\. Introduction

Kernel methods are well-established tools in machine learning which are
fundamental to support vector machine models for classification, nonlinear
regression and outlier detection involving small or moderate-sized data sets
[31], [5], [29]. Applications are manifold and include text classification
[21], protein classification [20] as well as applications to biological
sequences [35] and labelled graphs [17]. The essence of these methods is to
achieve better separation between labelled data by embedding a low-dimensional
feature space $X$ into a higher dimensional one $H$, which is commonly assumed
to be a Hilbert space, by means of a feature map $\psi:X\rightarrow H$. The
associated kernel is a function $K:X\times X\rightarrow\mathbb{R}$ with the
property that
$\left\langle\psi\left(x\right),\psi\left(y\right)\right\rangle_{H}=K\left(x,y\right)$
for all $x$ and $y$ in $X.$ If $K$ is known in closed form then the inner-
products of all extended features are obtainable from the evaluation of $K$ at
pairs of training instances in the original feature set. A typical
classification problem can be formulated as convex constrained optimisation
problem for which the Lagrangian dual involves only the inner-products of
pairs of enhanced features in the set of training instances. Crucially, one
does not need the vectors of the enhanced features themselves. This
observation – the basis of the so-called kernel trick – then allows one to
enjoy the advantages of working in a higher dimensional feature space without
some of the concomitant drawbacks.

The selection of an effective kernel is challenging and somewhat task-
dependent. When the training data consist of sequential data such as time
series, these challenges are magnified. To address these and other
difficulties much recent progress has been made by re-purposing the (path)
signature transform from rough path theory, which has decisive advantages in
capturing complex interactions between multivariate data streams. We recall
that the signature of a continuous bounded variation path
$\gamma:\left[a,b\right]\rightarrow V$ is the formal tensor series of iterated
integrals

(1.1)
$S\left(\gamma\right)_{a,b}=1+\sum_{k=1}^{\infty}S\left(\gamma\right)_{a,b}^{k}\in
T\left(\left(V\right)\right)=\oplus_{k=0}^{\infty}V^{\otimes k}\text{ with
}S\left(\gamma\right)_{a,b}^{k}:=\int_{a<t_{1}<t_{2}<...<t_{k}<b}d\gamma_{t_{1}}\otimes...\otimes
d\gamma_{t_{k}}.$

The soundness of this approach is underpinned by the fact that the map
$\gamma\mapsto S\left(\gamma\right)_{a,b}$ is one-to-one, up to an equivalence
relation on the space of paths [16]. The signature is invariant under
reparameterisation, and therefore by representing the path $\gamma$ by the
tensor series $S(\gamma)_{a,b}$ one removes an otherwise complicating
infinite-dimensional symmetry. On the other hand, the signature captures the
order of events along $\gamma.$ The algebraic properties of the signature have
been developed since the foundational work of Chen; it is now well understood
that the signature transform describes the set of polynomials on
unparameterised paths, in a sense that can be made meaningful. Analytically,
the signature of $\gamma$ characterises the class of responses (i.e.
solutions) of all smooth differential systems which have $\gamma$ as the
input.

An important fact is the factorial decay rate of the terms in the series in
(1.1). That is, given appropriately defined norms on the tensor product spaces
$V^{\otimes k}:$

$\left|\left|\int_{a<t_{1}<t_{2}<...<t_{k}<b}d\gamma_{t_{1}}\otimes...\otimes
d\gamma_{t_{k}}\right|\right|_{V^{\otimes
k}}\leq\frac{L\left(\gamma\right)^{k}}{k!},$

where $L\left(\gamma\right)$ denotes the length of the path over
$\left[a,b\right].$ This allows one to define the (untruncated) signature
kernel of two paths $\gamma$ and $\sigma$ by

(1.2) $K^{\gamma,\sigma}\left(s,t\right)=\left\langle
S\left(\gamma\right)_{a,s},S\left(\sigma\right)_{a,t}\right\rangle:=1+\sum_{k=1}^{\infty}\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}$

where $\left\langle\cdot,\cdot\right\rangle_{k}$ is the canonical (Hilbert-
Schmidt) inner-product on $V^{\otimes k}$ derived from a fixed inner-product
on $V$. In the recent paper [6] it was shown that this untruncated signature
kernel has some advantages over it truncated counterpart [18] which, in some
cases, lead to greater accuracy in classification and regression tasks on
benchmark data sets for multivariate time series. The explanation for this
turns on the key observation that $K$ is the unique solution of the hyperbolic
partial differential equation

(1.3) $\frac{\partial^{2}K}{\partial s\partial
t}\left(s,t\right)=K\left(s,t\right)\left\langle\gamma_{s}^{\prime},\sigma_{t}^{\prime}\right\rangle\text{
with }K\left(a,\cdot\right)=K\left(\cdot,a\right)\equiv 1.$

The solution to which can be approximated using PDE solvers, thus allowing for
the efficient computation of the inner product in (1.2) and obviating the need
to compute iterated integrals.

While the kernel (1.2) is useful, it is also in some respects confining. One
restriction it imposes is on the relative contributions made to the sum (1.2)
by the different inner-products $\left\langle\cdot,\cdot\right\rangle_{k}$. It
is easy to see for example by scaling $\gamma$ by
$\lambda=e^{\alpha}\in\mathbb{R}$ to give
$(\lambda\gamma)_{\cdot}=\lambda\gamma_{\cdot}$ we have

$K^{\lambda\gamma,\sigma}\left(s,t\right)=1+\sum_{k=1}^{\infty}e^{\alpha
k}\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k},$

so that the signature kernel for the family of inner-products
$\left\langle\cdot,\cdot\right\rangle_{\alpha}=\sum_{k\geq 0}e^{\alpha
k}\left\langle\cdot,\cdot\right\rangle_{k}$ can be obtained as above by
solving the appropriately rescaled version of the PDE (1.3). The starting
point for this paper is to introduce methods that allow for the efficient
computation of general signature kernels with a different weighting. These
will be derived from bilinear forms on $T\left(V\right)$ of the type

$\left\langle\cdot,\cdot\right\rangle_{\phi}=\sum_{k=0}^{\infty}\phi\left(k\right)\left\langle\cdot,\cdot\right\rangle_{k},$

where $\phi:\mathbb{N\cup}\left\\{0\right\\}\rightarrow\mathbb{R}$ (or,
sometimes, $\mathbb{C}$), so that
$\left\langle\cdot,\cdot\right\rangle_{\phi}$ need not even define an inner-
product. One fundamental observation we take advantage of is illustrated by
the following argument: assume $\phi\left(0\right)=1,$ and suppose that we can
solve the Hamburger moment problem for the sequence
$\left\\{\phi\left(k\right):k\in\mathbb{N\cup}\left\\{0\right\\}\right\\}$,
i.e. we can find a probability measure $\mu$ on $\mathbb{R}$ such that

(1.4) $\phi\left(k\right)=\int\lambda^{k}d\mu\left(\lambda\right)\text{ for
all }k\in\mathbb{N\cup}\left\\{0\right\\}.$

Then, under some conditions on $\mu,$ we will be able to justify the following
identity

(1.5) $\left\langle
S\left(\gamma\right)_{a,s},S\left(\sigma\right)_{a,t}\right\rangle_{\phi}=\sum_{k=0}^{\infty}\int\lambda^{k}\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}d\mu\left(\lambda\right)=\int
K^{\lambda\gamma,\sigma}\left(s,t\right)d\mu\left(\lambda\right).$

In this case, the computation of the $\phi-$signature kernel, i.e. the one
arising from $\left\langle\cdot,\cdot\right\rangle_{\phi},$ will amount to
integrating scaled solutions to the PDE (1.3) in $\lambda$ with respect to the
measure $\mu.$ The practicability of this approach depends on two aspects.
Firstly, one needs to be able to solve the moment problem (1.4); there are
well-known necessary and sufficient conditions but, ideally, $\mu$ should be
determined explicitly. Secondly, one needs to be able to approximate
accurately the integral on the right hand side of (1.5). In this respect one
is helped by the form of the function $\lambda\mapsto
K^{\lambda\gamma,\sigma}\left(s,t\right)$ which is real analytic with a power
series whose coefficients decay at rate $\left(n!\right)^{-2}$. Hence, in
cases where $\mu$ has a density $w$ given in closed form, Gaussian quadrature
provides an approximation of the form

$\int
K^{\lambda\gamma,\sigma}\left(s,t\right)d\mu\left(\lambda\right)\approx\sum_{i=1}^{m}w_{i}K^{\lambda_{i}\gamma,\sigma}\left(s,t\right)$

and equip us with well-described error bounds, see e.g. [32]. For these
examples, the $\phi-$signature kernel can be approximated at the expense of
$m$ implementations of a PDE solver.

The same principle outlined in the previous paragraph can appear in different
guises. For example, by solving the trigonometric moment problem

$\phi\left(k\right)=\int_{0}^{2\pi}e^{ik\theta}d\mu\left(\theta\right)\text{
for }k\in\mathbb{Z}$

to find a measure $\mu$ on $\left[0,2\pi\right]$, then an analogue of (1.5)
can be obtained by integrating the complex-valued function $\theta\mapsto$
$K^{\exp\left(i\theta\right)\gamma,\sigma}\left(s,t\right)$ with respect to
$\mu.$ A similar observation applies to a class of integral transforms having
the form

(1.6) $\phi\left(u\right)=\int_{C}r\left(u,z\right)d\mu\left(z\right);\text{
where }r\left(u,z\right)=g\left(z\right)^{\alpha u}\text{ }\in\mathbb{C}\text{
for some }\alpha\in\mathbb{R}.$

This class includes the Fourier-, Laplace- and Mellin- Stieltjes transforms,
for which specific pairs $\left(\phi,\mu\right)$ are of course extensively
documented. We illustrate a range of examples that can be generated using this
idea in the main text.

Extensions of the same idea apply to expected signatures. It is by now well
known that, under some conditions, the expected signature of a stochastic
process characterises the law of that process [7]. This motivates the use of
expected signatures as a measure of similarity of two laws on path space, for
example through the quantity

$d_{\phi}\left(\mu,\nu\right):=\left|\left|\mathbb{E}^{\mu}\left[S\left(X\right)\right]-\mathbb{E}^{\nu}\left[S\left(X\right)\right]\right|\right|_{\phi},$

which is seen to be a maximum mean discrepancy (MMD) distance between $\mu$
and $\nu$; see [15] and [8]. We also have a measure of alignment of the two
expected signatures of $\mu$ and $\nu$ given by

$\cos\angle_{\phi}\left(\mu,\nu\right):=\frac{\left\langle\mathbb{E}^{\mu}\left[S\left(X\right)\right],\mathbb{E}^{\nu}\left[S\left(X\right)\right]\right\rangle_{\phi}}{\left|\left|\mathbb{E}^{\mu}\left[S\left(X\right)\right]\right|\right|_{\phi}\left|\left|\mathbb{E}^{\nu}\left[S\left(X\right)\right]\right|\right|_{\phi}},$

which can be interpreted as an analogue of the Pearson correlation coefficient
for measures on path space. As an application we consider designing goodness-
of -fit tests in which one wants to understand when an observed empirical
sample is drawn from a well-described baseline distribution. A motivating
example for this paper was that of the detection of radio frequency
interference (RFI) contamination in radioastronomy. In this situation,
electrical signals are collected from an array of antennas [36]. Under the
null hypothesis of no RFI contamination, the signals will reflect only the so-
called thermal noise of the receiving equipment. From this perspective, the
most important reference distribution will that of white noise or, in its
integrated form, Brownian motion. Kernels have been used for similar problems
previously, albeit for the case of vector-valued data, see e.g. [9]. Proposals
have been made to put similar ideas in to practice in the context of two-sided
statistical tests determine whether two observed empirical measures on paths
are drawn from the same underlying distribution. For example [8] work using
the truncated signature kernel, while [19] present an application based on the
original signature kernel $\phi\equiv 1$.

A formula for the expected Stratonovich signature of multivariate Brownian
motion has been known since the work of Fawcett [14] and Victoir [25]. In the
context of the problems described above, we can take advantage of Fawcett’s
formula to prove what we believe to be a novel identity, namely that for any
continuous path $\gamma$ of bounded variation we have

(1.7) $\left\langle\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right],S\left(\gamma\right)_{0,t}\right\rangle_{\phi}=\cosh\left(\rho_{\sqrt{s/2}\gamma}\left(t\right)\right).$

In this formula, $\rho_{\gamma}\left(t\right)$ is the hyperbolic distance
between the starting point and the end point of the hyperbolic development of
the path segment $\left.\gamma\right|_{\left[0,t\right]}$, and

$\phi\left(k\right):=\Gamma\left(k/2+1\right):=\int_{0}^{\infty}x^{k/2}e^{-x}dx.$

When we realise hyperbolic space as a hyperboloid, the right hand side of
formula (1.7) can be obtained by solving a linear ordinary differential
equation. In the special case where $\gamma$ is piecewise linear, this
solution of the equation is a known product of matrices. These remarks allow
one to compute quantities like $d_{\phi}\left(\mathcal{W},\nu\right)$, where
$\mathcal{W}$ denotes Wiener measure and $\nu$ is an empirical measure on
bounded variation paths. We note that the primary use of the hyperbolic
development in the study of signatures to date has been in obtaining lower
bounds for the study of signature asymptotics, see [16] and [2]. In this
context, the identity (1.7) appears new, and it establishes a connection
between the signature kernel and these broader topics. It seems plausible that
an additional benefit of (1.7) will be that it allows a more analytic
treatment of these other problems in a way that relies less on the geometrical
intricacies of hyperbolic space.

If $\phi\equiv 1$, we can use Hankel’s well-known representation for the
reciprocal Gamma function as the contour integral

$\frac{1}{\Gamma\left(z\right)}=\frac{1}{2\pi i}\oint_{H}w^{-z}e^{w}dw,$

where $H$ is Hankel’s contour. Noting the similarity with (1.6) we can obtain
the identity

(1.8) $\left\langle\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right],S\left(\gamma\right)_{0,t}\right\rangle_{\phi}=\frac{1}{2\pi
i}\oint_{C}w^{-1}e^{w}\cosh\left(\rho_{\sqrt{s/2w}\gamma}\left(t\right)\right)dw,$

for an appropriate contour $C$. To make sense of this formula, we first need
to make sense of the complex rescaling in the defining ODE for hyperbolic
development. The numerical evaluation of contour integrals of the form
$\oint_{C}f\left(w\right)e^{w}dw$ is an active topic in numerical integration,
see [27], and we use these ideas to evaluate (1.8). The same idea can be
extended to cover general $\phi.$

In the final two sections we consider examples which lend themselves to being
treated by the methods outlined above. A natural question is how to select an
appropriate $\phi$ for a given task and, the related question of how to
evaluate the performance of a given kernel against an alternative. To develop
this, we reverse the perspective taken above and use $d_{\phi}$ to define a
loss function

$L_{\phi}\left(\mathcal{W},\mu\right):=d_{\phi}\left(\mathcal{W},\mu\right)_{\phi}^{2}$

and, given a finite collection of paths
$\left\\{\gamma_{1},...,\gamma_{n}\right\\},$ we consider the problem of
minimising $L$ over the set

$C_{n}=\left\\{\mu={\textstyle\sum\nolimits_{i=1}^{n}}\lambda_{i}\delta_{\gamma_{i}}:{\textstyle\sum\nolimits_{i=1}^{n}}\lambda_{i}=1,\lambda_{i}\geq
0\right\\}.$

Under some conditions on the support this optimisation problem will have a
unique solution $\mu^{\ast}$ which we can find. This gives us a way of
evaluating the similarity of a given finitely supported (possibly empirical)
measure $\mu$ to Wiener measure under the loss function induced by
$\left\langle\cdot,\cdot\right\rangle_{\phi}$ by comparing
$L_{\phi}\left(\mathcal{W},\mu\right)$ and
$L_{\phi}\left(\mathcal{W},\mu^{\ast}\right)$. For example, if the ratio
$\frac{L_{\phi}\left(\mathcal{W},\mu^{\ast}\right)}{L_{\phi}\left(\mathcal{W},\mu\right)}<\alpha<<1$
then by an appropriate selection of the threshold $\alpha$ one might decide
that $\mu$ does not resemble Wiener measure. We do not give an extensive
treatment of examples, but to illustrate how these methods introduced above
might be used we consider two cases in detail:

  1. (1)

Cubature measures of degree $N$ on Weiner space are finitely supported
measures which matched the expected iterated integrals of Brownian motion up
to and including degree $N.$ Explicit constructions are known in some cases,
see [25]. By definition these measures will be optimal in the above sense for
any kernel induced by any $\phi$ with $\phi\left(k\right)=0$ for $k\geq N.$
One might expect that they are close to optimal for smoother $\phi$ which
still decay sufficiently fast.

  2. (2)

We model radio frequency interference in sky-subtracted visibilities
radioastronomy as advocated by [36] and consider two idealised types of signal
contamination:

     * •

Narrow-band RFI measure across $n$ antennas. In this case the received signals
are $n$ linear superpositions of independent Brownian motions with a single-
frequency sinusoidal wave of a fixed amplitude.

     * •

Short duration high energy bursts. As a model for this we consider the
gerneralisation to the multivariate case of the example, originally considered
in the univariate setting in which the signal is given by
$X_{t}=W_{t}+\epsilon\sqrt{\left(t-U\right)_{+}}$ for
$t\in\left[0,1\right],$where $\left(W_{t}\right)_{t\in\left[0,1\right]}$ is a
Brownian motion, $U$ is independent an uniformly distributed on
$\left[0,1\right]$ and $\epsilon>0.$ The theoretical interest in this comes
from the existence of a critical parameter $\epsilon_{0}>0$ for which the law
of $X$ is equivalent to $\mathcal{W}$ if and only if $\epsilon<\epsilon_{0},$
see [10], and which therefore gives an example that falls outside the scope of
traditional maximum-likelihood-based approaches to the problem.

##  2\. Background on General Signature Kernels

Let $T\left(V\right)$ denote the algebra of tensor polynomials over a finite
dimensional vector space $V$ which consists of elements of the form

$a=\sum_{k=0}^{\infty}a_{k},\text{ }a_{k}\in V^{\otimes k}\text{ such that
}a_{k}=0\text{ for all but finitely many }k,$

with the tensor product defined by

$ab=\sum_{k=0}^{\infty}\sum_{l=0}^{k}a_{l}b_{k-l}\text{ }$

where the product $V^{\otimes l}\times
V^{\otimes\left(k-l\right)}\ni\left(c,d\right)\mapsto cd\in V^{\otimes k}$ is
determined by $\left((v_{1}...v_{l}),(v_{l+1}...v_{k})\right)\mapsto
v_{1}...v_{k}$ for $v_{1}...v_{k}\in V.$ We let $T\left(\left(V\right)\right)$
denote the space of formal tensor series, and $V^{\ast}$ denote the
(algebraic) dual space of $V.$ Then $T\left(V^{\ast}\right)$ is the dual space
of $T\left(\left(V\right)\right),$ and the signature of a continuous bounded
variation path $\gamma:[a,b]\rightarrow$ $V$ is the family of elements
$\\{S\left(\gamma\right)_{s,t}:s\leq t\in[a,b]\\}$ in
$T\left(\left(V\right)\right)$ determined inductively by

(2.1) $S\left(\gamma\right)_{s,t}\left(1\right)=1\text{ and
}S\left(\gamma\right)_{s,t}\left(v_{1}...v_{k}\right)=\int_{s}^{t}S\left(\gamma\right)_{s,u}\left(v_{1}...v_{k-1}\right)d\gamma_{u}\left(v_{k}\right),\text{
with }v_{1},...,v_{k}\in V^{\ast}.$

We will write

(2.2)
$S\left(\gamma\right)_{s,t}=1+\sum_{k=1}^{\infty}\int_{s<t_{1}<...<t_{k}<t}d\gamma_{t_{1}}...d\gamma_{t_{k}}\in
T\left(\left(V\right)\right),$

and let

(2.3) $\mathcal{S}=\\{S\left(\gamma\right)_{s,t}:\gamma,s<t\\}\subset
T\left(\left(V\right)\right).$

We consider dual pairs $\left(E,F\right),\,$where $E\ \ $and $F$ are two
linear subspaces of $T\left(\left(V\right)\right).$ Recall that this means
that $\left(\cdot,\cdot\right):E\times F\rightarrow\mathbb{R}$ is a bilinear
map such that the linear functionals $\left\\{\left(e,\cdot\right):e\in
E\right\\}\subset F^{\ast}$ and $\left\\{\left(\cdot,f\right):f\in
F\right\\}\subset E^{\ast}$ separate points in $F$ and $E$ respectively. We
can identify $E$ and $F$ linear subspaces of the algebraic dual spaces
$F^{\ast}$ and $E^{\ast}$ respectively.

######  Definition 2.1.

Let $\left(E,F\right)$ be a dual pair as above. Suppose that
$\mathcal{S}\subset E\cap F$ where $S$ denotes the set of signatures (2.3).
Then given two continuous paths $\gamma,\sigma:[a,b]\rightarrow V$ of bounded
variation, we define the $\left(\cdot,\cdot\right)$-signature kernel of
$\gamma$ and $\sigma$ to be the function

$\left[a,b\right]\times\left[a,b\right]\ni\left(s,t\right)\mapsto\left(S\left(\gamma\right)_{a,s},S\left(\sigma\right)_{a,t}\right)=K_{\left(\cdot,\cdot\right)}^{\gamma,\sigma}\left(s,t\right).$

######  Remark 2.2.

This definition is not symmetric in general, i.e. it may hold that
$K_{\left(\cdot,\cdot\right)}^{\gamma,\sigma}\neq$
$K_{\left(\cdot,\cdot\right)}^{\sigma,\gamma}.$

For this definition to be useful we need to demand more of the pairing
$\left(E,F\right).$ More exactly we need at least that their continuous duals
satisfy $F\subseteq E^{\prime}$ and $E\subseteq F^{\prime}.$ To go further
still we will need that they respect some of the algebraic structure on
$T\left(\left(V\right)\right).$ The examples we will work are derived from a
fixed but arbitrary inner product $\left\langle\cdot,\cdot\right\rangle$ on
$V.$ This gives rise to the Hilbert-Schmidt inner product
$\left\langle\cdot,\cdot\right\rangle_{k}$ on the $k$-fold tensor product
spaces $V^{\otimes k}$ in a canonical way. Then, by taking

$\left\langle
a,b\right\rangle_{\phi}:=\sum_{k=0}^{\infty}\phi\left(k\right)\left\langle
a_{k},b_{k}\right\rangle_{k}$

for some weight function
$\phi:\mathbb{N}\cup\left\\{0\right\\}\rightarrow\mathbb{R}_{+}$ we may define
$T_{\phi}\left(V\right)$ to be the Hilbert space obtained by completing
$T\left(V\right)$ with respect to
$\left\langle\cdot,\cdot\right\rangle_{\phi}.$ We equip
$T_{\phi}\left(V\right)$ with the norm topology unless stated otherwise. It is
necessary to have a condition on $\phi$ which ensures that
$\mathcal{S\subset}T_{\phi}\left(V\right).$

######  Lemma 2.3.

Let $\phi:\mathbb{N}\cup\left\\{0\right\\}\rightarrow\mathbb{R}_{+}$ be such
that for every $C>0$ the series
$\sum_{k\in\mathbb{N}}C^{k}\phi\left(k\right)\left(k!\right)^{-2}$ is
summable. Then $\mathcal{S\subset}T_{\phi}\left(V\right).$

###### Proof.

Let $\left\\{e_{i}:i=1,..,d\right\\}$ be any orthonormal basis of $V$ w.r.t.
$\left\langle\cdot,\cdot\right\rangle,$ and
$\left\\{e_{I}^{\ast}:I=(i_{1},...,i_{k})\right\\}$ the associated dual basis
on $\left(V^{\ast}\right)^{\otimes k}.$ Then

(2.4)
$\left|\left|S\left(\gamma\right)_{s,t}\right|\right|_{\phi}^{2}=\sum_{k=0}^{\infty}\phi\left(k\right)\sum_{\left|I\right|=k}\left[S\left(\gamma\right)_{s,t}\left(e_{I}^{\ast}\right)\right]^{2},$

and since

$S\left(\gamma\right)_{s,t}\left(e_{I}^{\ast}\right)=\int_{s<u_{1}<u_{2}<...<u_{k}<t}d\left\langle
e_{i_{1}},\gamma_{u_{1}}\right\rangle d\left\langle
e_{i_{2}},\gamma_{u_{2}}\right\rangle...d\left\langle
e_{i_{k}},\gamma_{u}\right\rangle$

we can estimate the summands in (2.4) by

$\sum_{\left|I\right|=k}\left[S\left(\gamma\right)_{s,t}\left(e_{I}^{*}\right)\right]^{2}\leq\frac{L_{s,t}\left(\gamma\right)^{2k}}{\left(k!\right)^{2}},\text{
where }L_{s,t}\left(\gamma\right):=\int_{s}^{t}\left|d\gamma_{u}\right|\text{
is the length of }\gamma.$

The summability condition then ensures that (2.4) is finite. ∎

This prompts the following condition.

######  Condition 1.

The function $\phi:\mathbb{N}\cup\left\\{0\right\\}\rightarrow\mathbb{R}_{+}$
is such that the series
$\sum_{k\in\mathbb{N}}C^{k}\phi\left(k\right)\left(k!\right)^{-2}$ is summable
for every $C>0.$

The next lemma describes examples of dual pairs $\left(E,F\right)$ of Hilbert
spaces which fulfill the conditions in Definition 2.1.

######  Lemma 2.4.

Let $\phi:\mathbb{N}\cup\left\\{0\right\\}\rightarrow\mathbb{R}_{+}$ and
$\psi:\mathbb{N}\cup\left\\{0\right\\}\rightarrow\mathbb{R}_{+}$ be functions
such that $\phi$ and $\psi^{-1}($i.e. $n\mapsto\psi\left(n\right)^{-1})$
satisfy the summability criterion of Condition 1. In each of the following
cases $\left(E,F\right)$ is a dual pair which satisfies $F\subseteq
E^{\prime}$ and $E\subseteq F^{\prime}.$

  1. (1)

$E=T_{\phi}\left(V\right),F=T_{\phi}\left(V\right),\left(\cdot,\cdot\right)=\left\langle\cdot,\cdot\right\rangle_{\phi},$

  2. (2)

$E=T_{\phi}\left(V\right),F=T_{\psi^{-1}}\left(V\right),\left(\cdot,\cdot\right)=\left\langle\cdot,\cdot\right\rangle_{\sqrt{\phi/\psi}}$.

###### Proof.

For notational ease we write $H_{\phi}$ for $T_{\phi}\left(V\right).$ In both
cases Condition 1 ensures that $\mathcal{S}\subset E\cap F.$ In case 1, it is
classical that $H_{\phi}^{\prime}=\\{\left\langle
h,\cdot\right\rangle_{\phi}:h\in H_{\phi}\\},\,\ $while for case 2 we have for
$h\in H_{\phi}$ and $g\in H_{\psi^{-1}}$ we have that

$\left|\left\langle
h,g\right\rangle_{\sqrt{\phi/\psi}}\right|=\left|\sum_{k=0}^{\infty}\sqrt{\frac{\phi\left(k\right)}{\psi\left(k\right)}}\left\langle
h_{k},g_{k}\right\rangle_{k}\right|\leq\left|\left|h\right|\right|_{\phi}\left|\left|g\right|\right|_{\psi^{-1}}$

hence $\\{\left\langle h,\cdot\right\rangle_{\sqrt{\phi/\psi}}:h\in
H_{\phi}\\}\subseteq H_{\psi^{-1}}^{\prime}$. By using the fact that $h$ is in
$H_{\phi}$ if and only if
$\tilde{h}:=\sqrt{\phi\psi}h:=\sum_{k}\sqrt{\phi\left(k\right)\psi\left(k\right)}h_{k}$
is in $H_{\psi^{-1}}$ we see that

$\left\langle
h,\cdot\right\rangle_{\sqrt{\phi/\psi}}=\left\langle\tilde{h},\cdot\right\rangle_{\psi^{-1}}$

so that $\\{\left\langle h,\cdot\right\rangle_{\sqrt{\phi/\psi}}:h\in
H_{\phi}\\}=\\{\left\langle h,\cdot\right\rangle_{\psi^{-1}}:h\in
H_{\psi^{-1}}\\}=H_{\psi^{-1}}^{\prime}.$ ∎

Hereafter we will work almost entirely in the case $\left\langle
T_{\phi}\left(V\right),T_{\phi}\left(V\right)\right\rangle_{\phi}$ in which
the dual pair is the Hilbert space $T_{\phi}\left(V\right)$ with itself with
pairing given by the inner product
$\left\langle\cdot,\cdot\right\rangle_{\phi}.$ This leads to the following
definition.

######  Definition 2.5.

Let $\phi:\mathbb{N}\cup\left\\{0\right\\}\rightarrow\mathbb{R}_{+}$ satisfy
Condition 1. Given two continuous paths $\gamma,\sigma:[a,b]\rightarrow V$ of
bounded variation, we define the $\phi$-signature kernel of $\gamma$ and
$\sigma$ to be the two-parameter function $K_{\phi}^{\gamma,\sigma}$ defined
by

$\left[a,b\right]\times\left[a,b\right]\ni\left(s,t\right)\mapsto\left\langle
S\left(\gamma\right)_{a,s},S\left(\sigma\right)_{a,t}\right\rangle_{\phi}=:K_{\phi}^{\gamma,\sigma}\left(s,t\right).$

######  Remark 2.6.

It is straight forward to extend the discussion above to consider general
bilinear forms of signatures. If
$\phi:\mathbb{N}\cup\left\\{0\right\\}\rightarrow\mathbb{R}$, then we can
define a semi-definite inner product on $T\left(V\right)$ by

$\left\langle
a,b\right\rangle_{\left|\phi\right|}:=\sum_{k=0}^{\infty}|\phi\left(k\right)|\left\langle
a_{k},b_{k}\right\rangle_{k}.$

Let $N$ denote the linear subspace of $T\left(V\right)$ given by the kernel of
semi-norm $\left|\left|\cdot\right|\right|_{\left|\phi\right|}.$ Then we we
can complete the quotient space $T\left(V\right)/N$ with respect to inner
product $\left\langle\cdot,\cdot\right\rangle_{\left|\phi\right|}$ and denote
the resulting Hilbert space by $T_{|\phi|}\left(V\right).$ The bilinear form
on $T\left(V\right)$

(2.5) $\left\langle
a,b\right\rangle_{\phi}:=B_{\phi}\left(a,b\right):=\sum_{k=0}^{\infty}\phi\left(k\right)\left\langle
a_{k},b_{k}\right\rangle_{k}$

extends to a continuous bilinear form on $T_{|\phi|}\left(V\right)$. If $\phi$
is such that $\left|\phi\right|$ satisfies Condition 1 then, as above, we
define the $\phi$-signature kernel of $\gamma$ and $\sigma$ to be the function
$K_{\phi}^{\gamma,\sigma}:[a,b]\times\left[a,b\right]\rightarrow\mathbb{R}$ by

$K_{\phi}^{\gamma,\sigma}\left(s,t\right):=\left\langle
S\left(\gamma\right)_{a,s},S\left(\sigma\right)_{a,t}\right\rangle_{\phi}.$

This agrees with the previous definition whenever $\phi$ takes positive
values.

The following shifted weight functions arise naturally when doing calculus on
signature kernels.

######  Definition 2.7.

Given a function $\phi:\mathbb{N}\cup\left\\{0\right\\}\rightarrow\mathbb{R}$
and $k\in\mathbb{N}$, we define the $k$-shift of $\phi$ to be the function
$\phi_{+k}:\mathbb{N}\cup\left\\{0\right\\}\to\mathbb{R}$ determined by
$\phi_{+k}\left(\cdot\right)=\phi\left(\cdot+k\right).$

The next result is fundamental.

######  Proposition 2.8.

Let $\gamma,\sigma:[a,b]\rightarrow V$ be two continuous paths of bounded
variation. Assume that the function
$\phi:\mathbb{N}\cup\left\\{0\right\\}\rightarrow\mathbb{R}$ is such that
$\left|\phi\right|$ and its $1$-shift $\left|\phi_{+1}\right|$ both satisfy
Condition 1. Then the $\phi$\- and $\phi_{+1}$\- signature kernels of $\gamma$
and $\sigma$ are well defined and are related by the two-parameter integral
equation

$K_{\phi}^{\gamma,\sigma}\left(s,t\right)=\phi\left(0\right)+\int_{a}^{s}\int_{a}^{t}K_{\phi_{+1}}^{\gamma,\sigma}\left(u,v\right)\left\langle
d\gamma_{u},d\sigma_{v}\right\rangle.$

###### Proof.

Well definedness of the two signature kernels follows from the summability
conditions. Unravelling the definitions and using (2.1) gives

$\displaystyle K_{\phi}^{\gamma,\sigma}\left(s,t\right)$
$\displaystyle=\sum_{k=0}^{\infty}\phi\left(k\right)\sum_{\left|I\right|=k}S\left(\gamma\right)_{a,s}\left(e_{I}\right)S\left(\sigma\right)_{a,t}\left(e_{I}^{\ast}\right)$
$\displaystyle=\phi\left(0\right)+\sum_{k=1}^{\infty}\phi\left(k\right)\sum_{\left|I\right|=k-1}\int_{a}^{s}\int_{a}^{t}S\left(\gamma\right)_{a,u}\left(e_{I}^{\ast}\right)S\left(\sigma\right)_{a,v}\left(e_{I}^{\ast}\right)\left\langle
d\gamma_{u},d\sigma_{v}\right\rangle$
$\displaystyle=\phi\left(0\right)+\int_{a}^{s}\int_{a}^{t}\sum_{k=0}^{\infty}\phi\left(k+1\right)\sum_{\left|I\right|=k}S\left(\gamma\right)_{a,u}\left(e_{I}^{\ast}\right)S\left(\sigma\right)_{a,v}\left(e_{I}^{\ast}\right)\left\langle
d\gamma_{u},d\sigma_{v}\right\rangle$
$\displaystyle=\phi\left(0\right)+\int_{a}^{s}\int_{a}^{t}K_{\phi_{+1}}^{\gamma,\sigma}\left(s,t\right)\left\langle
d\gamma_{u},d\sigma_{v}\right\rangle.$

∎

In the special case where $\phi$ is constant we see that the shift
$\phi_{+k}=\phi$ for every $k$ and therefore $K_{\phi}^{\gamma,\sigma}$
satisfies

$K_{\phi}^{\gamma,\sigma}\left(s,t\right)=\phi\left(0\right)+\int_{a}^{s}\int_{a}^{t}K_{\phi}^{\gamma,\sigma}\left(s,t\right)\left\langle
d\gamma_{u},d\sigma_{v}\right\rangle,$

and in particular if $\gamma$ and $\sigma$ are differentiable and $\phi\equiv
1$ then we write $K_{\phi}^{\gamma,\sigma}=K^{\gamma,\sigma}$ and refer to it
as the original signature kernel. As was first shown in [6], it solves the
partial differential equation

(2.6) $\frac{\partial^{2}K^{\gamma,\sigma}\left(s,t\right)}{\partial s\partial
t}=K^{\gamma,\sigma}\left(s,t\right)\left\langle\gamma_{s}^{\prime},\sigma_{t}^{\prime}\right\rangle\text{
on }\left[a,b\right]\times\left[a,b\right]$

with boundary conditions $K\left(a,\cdot\right)\equiv
K\left(\cdot,a\right)\equiv 1.$ The same paper shows how the solution to (2.6)
can be approximated numerically, and how the methodology extends to the case
of rough paths. The approximate solution can then be used to implement kernel
learning methods for classification or regression tasks based on time series
as mentioned in the introduction, see [8, 18].

It is self-evident from Proposition 2.8 that for general $\phi$ the function
will not solve a PDE of the type (2.6). Nevertheless we can produce examples
of different $\phi$ which do by varying the inner-product
$\left\langle\cdot,\cdot\right\rangle$ on the underlying vector space $V,$ or
by scaling the inner product on $T\left(V\right)$ homogeneously with respect
the grading on $T(V)$. By the latter idea we mean that, for
$\theta\in\mathbb{R}$ we can define
$\delta_{\theta}:T\left(V\right)\rightarrow T\left(V\right)$ to be the unique
algebra homomorphism which is determined by scalar multiplication by $\theta$
on $V$ (i.e. $V\ni a\mapsto\theta a$), then we have

(2.7) $\delta_{\theta}a=\sum_{k=0}^{\infty}\theta^{k}a_{k},\,\ \,\text{if
}a=\sum_{k=0}^{\infty}a_{k}\in T\left(V\right).$

The following lemma explores the properties of $\delta_{\theta}$ when it is
extended to a homogeneous linear map defined on (a subspace of) the Hilbert
space $T_{\phi}\left(V\right).$

######  Lemma 2.9.

Suppose $0\neq\theta\in\mathbb{R}$ and let
$\phi:\mathbb{N}\cup\left\\{0\right\\}\rightarrow\mathbb{R}.$ Let
$\theta\phi:\mathbb{N}\cup\left\\{0\right\\}\rightarrow\mathbb{R}$ denote the
function defined by the pointwise product
$\left(\theta\phi\right)\left(n\right)=\theta^{n}\phi\left(n\right)$ and let
$\delta_{\theta}:T\left(V\right)\rightarrow T\left(V\right)$ be the linear
operator defined by (2.7). Then:

  1. (1)

For every $a,b\in T\left(V\right)$ we have the identity

(2.8) $\left\langle\delta_{\theta}a,b\right\rangle_{\phi}=\left\langle
a,\delta_{\theta}b\right\rangle_{\phi}=\left\langle
a,b\right\rangle_{\theta\phi},$

which extends to $a,b\in T_{\left|\theta\phi\right|}\left(V\right)$. The map
$\delta_{\theta}$ extends uniquely to an isomorphism between the Hilbert
spaces $T_{\theta^{2}|\phi|}\left(V\right)$ and $T_{|\phi|}\left(V\right);$

  2. (2)

For $\left|\theta\right|\leq 1\,$ and $\phi>0$ we have
$T_{\phi}\left(V\right)\subseteq T_{\theta^{2}\phi}\left(V\right)$ and
$\delta_{\theta}:T_{\phi}\left(V\right)$ $\rightarrow T_{\phi}\left(V\right)$
is a bounded self-adjoint linear 8operator with operator norm
$\left|\left|\delta_{\theta}\right|\right|\leq 1;$

  3. (3)

For $\left|\theta\right|>1$ and $\phi>0,\,\ \delta_{\theta}$ is a linear
operator $\delta_{\theta}:D(\delta_{\theta})$ $\rightarrow
T_{\phi}\left(V\right)$ with domain $T_{\theta^{2}\phi}\left(V\right)\subseteq
D(\delta_{\theta})\subset T_{\phi}\left(V\right).$ If furthermore $\phi$
satisfies Condition 1, then $D(\delta_{\theta})$ is dense in
$T_{\phi}\left(V\right)$ and $\delta_{\theta}$ is self-adjoint.

###### Proof.

For item 1, the identity (2.8) follows from (2.5). The extension to the
completion follows from the fact that
$|\left\langle\delta_{\theta}a,b\right\rangle_{\phi}|\leq\left|\left|a\right|\right|_{\left|\theta\phi\right|}\left|\left|b\right|\right|_{\left|\theta\phi\right|}.$That
$\delta_{\theta}$ is an isometry between the pre-Hilbert spaces
$(T\left(V\right)/N,\left\langle\cdot,\cdot\right\rangle_{\theta^{2}\left|\phi\right|})$
and
$(T\left(V\right)/N,\left\langle\cdot,\cdot\right\rangle_{\left|\phi\right|})$
follows from (2.8) and the identity $\delta_{\theta}^{2}=\delta_{\theta^{2}}$:

$\left\langle\delta_{\theta}a,\delta_{\theta}b\right\rangle_{\left|\phi\right|}=\left\langle
a,\delta_{\theta}^{2}b\right\rangle_{\left|\phi\right|}=\left\langle
a,b\right\rangle_{\theta^{2}\left|\phi\right|},$

which extends to the completion $T_{\theta^{2}|\phi|}\left(V\right)$.
Surjectivity follows from the fact that
$\delta_{\theta}\left(T\left(V\right)\right)=T\left(V\right)$ for any non-zero
$\theta.$ For item 2, it is readily seen that
$\left|\left|a\right|\right|_{\theta^{2}\phi}\leq\left|\left|a\right|\right|_{\phi}$
when $\left|\theta\right|\leq 1$ for all $a\in T\left(V\right)$ and hence that
$T_{\phi}\left(V\right)\subseteq T_{\theta^{2}\phi}\left(V\right).$ By item 1
we then have
$\left|\left|\delta_{\theta}a\right|\right|_{\phi}\leq\left|\left|a\right|\right|_{\phi}$
which then extends to $T_{\phi}\left(V\right).$ Self-adjointness follows from
the identity

(2.9)
$\left\langle\delta_{\theta}a,b\right\rangle_{\phi}=\sum_{k=0}^{\infty}\theta^{k}\phi\left(k\right)\left\langle
a_{k},b_{k}\right\rangle_{k}=\left\langle
a,\delta_{\theta}b\right\rangle_{\phi},\text{ for all }a,b\in
T_{\phi}\left(V\right).$

Finally, for item 3 we observe that $T_{\theta^{2}\phi}\left(V\right)$ is a
linear subspace of $T_{\phi}\left(V\right)$ and then that
$\delta_{\theta}(T_{\theta^{2}\phi}\left(V\right))\subseteq$
$T_{\phi}\left(V\right)$ using item 1. If $\phi$ satisfies Condition 1 then
the domain of $\delta_{\theta}$ contains the linear span of the set of
signatures $\mathcal{S}$ (recall (2.3)) which is dense in
$T_{\phi}\left(V\right)$. Self-adjointness is again a consequence of (2.9). ∎

As an immediate corollary we obtain the following result, which we shall use
repeatedly.

######  Corollary 2.10.

Suppose $\theta\in\mathbb{R}$ and let
$\phi:\mathbb{N}\cup\left\\{0\right\\}\rightarrow\mathbb{R}$ be such that
$\left|\phi\right|$ satisfies Condition 1 then

$K_{\theta\phi}^{\gamma,\sigma}\left(s,t\right)=K_{\phi}^{\theta\gamma,\sigma}\left(s,t\right)=K_{\phi}^{\gamma,\theta\sigma}\left(s,t\right)$

for every $\left(s,t\right)\in\left[a,b\right]\times\left[a,b\right],$ where
$\theta\gamma$ and $\theta\sigma$ denote the paths obtained by the pointwise
multiplication of $\theta$ with $\gamma$ and $\sigma$ respectively. In
particular if $\phi\equiv 1$ then $K_{\theta\phi}^{\gamma,\sigma}:=$
$K_{\theta}^{\gamma,\sigma}$ satisfies

$K_{\theta}^{\gamma,\sigma}\left(s,t\right)=1+\theta\int_{a}^{s}\int_{a}^{t}K_{\theta}^{\gamma,\sigma}\left(s,t\right)\left\langle
d\gamma_{u},d\sigma_{v}\right\rangle.$

###### Proof.

We use the fact that
$\delta_{\theta}S\left(\gamma\right)_{s,t}=S\left(\theta\gamma\right)_{s,t}$
and the previous lemma to observe that

$K_{\theta\phi}^{\gamma,\sigma}\left(s,t\right)=\left\langle
S\left(\gamma\right)_{a,s},S\left(\sigma\right)_{a,t}\right\rangle_{\theta\phi}=\left\langle\delta_{\theta}S\left(\gamma\right)_{a,s},S\left(\sigma\right)_{a,t}\right\rangle_{\phi}=K_{\phi}^{\theta\gamma,\sigma}\left(s,t\right).$

The fact that
$K_{\phi}^{\theta\gamma,\sigma}\left(s,t\right)=K_{\phi}^{\gamma,\theta\sigma}\left(s,t\right)$
follows from the self-adjointness of $\delta_{\theta}.$ ∎

##  3\. Representing General Signature Kernels

Let $\gamma$ be a continuous $V$-valued path of bounded variation. Under the
condition of Lemma 2.3, we can identify the signature
$S\left(\gamma\right)_{s,t}$ with an element of $T_{\phi}\left(V\right)$ and
we can write

$S\left(\gamma\right)_{s,t}:=\sum_{k=0}^{\infty}S\left(\gamma\right)_{s,t}^{k}\in
T_{\phi}\left(V\right),$

where

$S(\gamma)_{s,t}^{k}:=\int_{s<u_{1}<\cdots<u_{k}<t}d\gamma_{u_{1}}\cdots
d\gamma_{u_{k}}\in V^{\otimes k},\ k\geq 1$

and $S(\gamma)_{s,t}^{0}:=S(\gamma)_{s,t}(1)\equiv 1$.

Two properties in particular of the signature render it a good feature map.
First is its universality property; that is, provided one is careful about
definitions and topologies, continuous function on compact subspaces of paths
are uniformly approximable by linear functionals of the signature. Central to
this is a combination of the Stone-Weierstrass theorem and the identity

$S\left(\gamma\right)_{s,t}\left(f\right)S\left(\gamma\right)_{s,t}\left(g\right)=S\left(\gamma\right)_{s,t}\left(f\shuffle
g\right)\text{ for }f,g\in T\left(V^{\ast}\right),$

where $f\shuffle g\in T\left(V^{\ast}\right)$ denotes the shuffle product of
the linear functionals $f$ and $g,$ see [23]. The second property is that
signatures are characteristic in the sense that the expected signature of a
path-valued random variable will, under certain conditions, characterise the
law of that random variable, see [16, 18] for more details.

In the previous section we introduced the definition of the $\phi$-signature
kernel of continuous paths $\gamma$ and $\sigma$ to be the function
$K_{\phi}^{\gamma,\sigma}\left(s,t\right)$ . This amounts to reweighting the
terms in the signature to give more or less emphasis to high order terms
compared to the original signature kernel, i.e.
$\left\langle\cdot,\cdot\right\rangle_{\phi}$ for $\phi\equiv 1$. In the
present section, we will build an approach to representing $\phi$-signature
kernels in such a way that allows for efficient computation. The same idea is
presented in multiple guises and then specialised within each case to yield
particular examples. Before we present this method for $\phi$-signature
kernels, we consider the error estimates which arise using a naive truncation-
based approach.

###  3.1. Truncated Signature Kernels

In this subsection, we give an error estimate of the truncated
$\phi$-signature kernel and the full $\phi$-signature kernel of two continuous
bounded variation paths.

Let the truncated signature kernel be denoted

(3.1) $K_{\phi}^{(N)}(s,t):=\sum_{k=0}^{N}\phi(k)\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}=\sum_{k=0}^{N}\phi\left(k\right)\sum_{\left|I\right|=k}S\left(\gamma\right)_{a,s}\left(e_{I}^{*}\right)S\left(\sigma\right)_{a,t}\left(e_{I}^{*}\right).$

We have the following proposition.

######  Proposition 3.1.

Let $\gamma,\sigma:[a,b]\rightarrow V$ be two continuous paths of bounded
variation. Assume that the function
$\phi:\mathbb{N}\cup\left\\{0\right\\}\rightarrow\mathbb{R}$ is such that
$|\phi|$ satisfies Condition 1, then the truncated signature kernel
$K_{\phi}^{(N)}(s,t)$ converges to the $\phi$-signature kernel
$K_{\phi}^{\gamma,\sigma}(s,t)$ when $N$ goes to infinity, and the error bound
is

(3.2)
$\left|K_{\phi}^{\gamma,\sigma}(s,t)-K_{\phi}^{(N)}(s,t)\right|\leq\sum_{k=N+1}^{\infty}|\phi(k)|(L_{s}(\gamma)L_{t}(\sigma))^{k}(k!)^{-2}$

where $L_{s}(\gamma)$ is the length of the path segment $\gamma|_{[a,s]}$.

###### Proof.

By the Cauchy-Schwarz inequality, we have

$\displaystyle\left|K_{\phi}^{\gamma,\sigma}(s,t)-K_{\phi}^{(N)}(s,t)\right|$
$\displaystyle\leq\sum_{k=N+1}^{\infty}|\phi(k)|\left|\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}\right|$
$\displaystyle\leq\sum_{k=N+1}^{\infty}|\phi(k)|\left|\left|S\left(\gamma\right)_{a,s}^{k}\right|\right|_{k}\left|\left|S\left(\sigma\right)_{a,t}^{k}\right|\right|_{k}$
$\displaystyle=\sum_{k=N+1}^{\infty}|\phi(k)|\frac{(L_{s}(\gamma)L_{t}(\sigma))^{k}}{(k!)^{2}}.$

Since $|\phi|$ satisfies Condition 1, the error goes to 0 as $N\to\infty$. ∎

We analyse two concrete examples that we will revisit later using other
methods.

  * •

The first example takes $\phi$ to be

(3.3) $\phi(k):=\left(\frac{k}{2}\right)!:=\Gamma\left(\frac{k}{2}+1\right)$

which plays an important role in Section 5 when we consider the expected
signature of Brownian motion.

  * •

The second example is

(3.4) $\phi(k)=\frac{\Gamma(m+1)\Gamma(k+1)}{\Gamma(k+m+1)}$

where $m\in\mathbb{R}_{+}$. The case when $m=0$, $\phi(k)\equiv 1$ corresponds
to the original signature kernel, while $m=1$ gives $\phi(k)=\frac{1}{k+1}$
which are the sequence of moments of a random variable which is uniformly
distributed on $[0,1]$.

The following corollary specialises the previously-obtained error estimate to
these cases.

######  Corollary 3.2.

Let $\gamma,\sigma:[a,b]\rightarrow V$ be two continuous paths of bounded
variation. Denote the length of the path segment $\gamma|_{[a,s]}$ as
$L_{s}(\gamma)$.

(1) The $\phi$-signature kernel under $\phi(k)=\left(\frac{k}{2}\right)!$ is
well defined and there is a constant $C$ such that

(3.5) $\left|K_{\phi}^{\gamma,\sigma}(s,t)-K_{\phi}^{(N)}(s,t)\right|\leq
C\left(\frac{e}{2N+2}\right)^{N+1/2}e_{N+1}\left(L_{s}(\gamma)L_{t}(\sigma)\right)$

where $e_{N+1}(x):=\sum_{k=N+1}^{\infty}\frac{x^{k}}{k!}$.

(2) The $\phi$-signature kernel under
$\phi(k)=\frac{\Gamma(m+1)\Gamma(k+1)}{\Gamma(k+m+1)}$ is well defined and the
error bound is

(3.6)
$\left|K_{\phi}^{\gamma,\sigma}(s,t)-K_{\phi}^{(N)}(s,t)\right|\leq\frac{\Gamma(m+1)}{\left(L_{s}(\gamma)L_{t}(\sigma)\right)^{\frac{m}{2}}}I_{m}^{(N+1)}\left(2\sqrt{L_{s}(\gamma)L_{t}(\sigma)}\right)$

in which
$I_{m}^{(N+1)}\left(z\right):=\left(\frac{z}{2}\right)^{m}\sum_{k=N+1}^{\infty}\frac{\left(\frac{1}{4}z^{2}\right)^{k}}{\Gamma(k+m+1)\Gamma(k+1)}$
is the tail of the series defining the modified Bessel function
$I_{m}\left(z\right)$ of the first kind of order $m$.

###### Proof.

It is easy to see that these two functions $\phi$ satisfy Condition 1, which
makes sure that the $\phi$-signature kernels are well defined. For the error
bound (3.5), by the Stirling’s approximation, there exist two constants
$C_{1},C_{2}$ such that

$C_{1}x^{x+\frac{1}{2}}e^{-x}\leq x!\leq C_{2}x^{x+\frac{1}{2}}e^{-x},\
\forall\ x>0.$

Then we have

$\frac{\left(\frac{k}{2}\right)!}{k!}\leq\frac{C_{2}}{\sqrt{2}C_{1}}\left(\frac{e}{2k}\right)^{\frac{k}{2}}$

and the sequence on the right hand side is decreasing. Let
$C=\frac{C_{2}}{\sqrt{2}C_{1}}$ and combine Proposition 3.1, it is easy to
show the error bound (3.5).

For the error bound (3.6), since the modified Bessel function
$I_{m}\left(2\sqrt{L_{s}(\gamma)L_{t}(\sigma)}\right)$ of the first kind of
order $m$ is defined by the series

$I_{m}\left(2\sqrt{L_{s}(\gamma)L_{t}(\sigma)}\right)=\left(L_{s}(\gamma)L_{t}(\sigma)\right)^{\frac{m}{2}}\sum_{k=0}^{\infty}\frac{(L_{s}(\gamma)L_{t}(\sigma))^{k}}{\Gamma(k+m+1)\Gamma(k+1)},$

the error bound follows from Proposition 3.1. ∎

###  3.2. General Signature Kernels by Randomisation

We now show how $\phi$-signature kernels can be represented, under suitable
integrability conditions, as the average of rescaled PDE solutions whenever
the sequence $\\{\phi(k):k=0,1,...\\}$ coincides with the sequence of moments
of a random variable. This representation consolidates the connection between
the original and the $\phi$-signature kernels in these cases. The connection
is captured in the following result.

######  Proposition 3.3.

Suppose $\pi$ is a random variable with finite moments of all orders and let
the functions

(3.7) $\phi(k)=\mathbb{E}[\pi^{k}]\ \text{and}\
\psi(k)=\mathbb{E}[|\pi|^{k}],\ \forall k\geq 0.$

We assume that $\psi$ satisfies Condition 1. Then the $\phi$-signature kernel
$K_{\phi}^{\gamma,\sigma}(s,t)$ of continuous bounded variation paths $\gamma$
and $\sigma$ is well defined and

(3.8)
$K_{\phi}^{\gamma,\sigma}(s,t)=\mathbb{E}_{\pi}\left[K^{\pi\gamma,\sigma}(s,t)\right]=\mathbb{E}_{\pi}\left[K^{\gamma,\pi\sigma}(s,t)\right].$

###### Proof.

Since $|\phi|$ satisfies Condition 1, which follows from the condition of
$\psi$, the $\phi$-signature kernel $K_{\phi}^{\gamma,\sigma}(s,t)$ is well
defined. Furthermore, $\psi$ satisfies Condition 1, by Fubini theorem, we have

$\displaystyle K_{\phi}^{\gamma,\sigma}(s,t)$
$\displaystyle=\sum_{k=0}^{\infty}\mathbb{E}\left[\pi^{k}\right]\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}$
$\displaystyle=\mathbb{E}\left[\sum_{k=0}^{\infty}\pi^{k}\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}\right]$
$\displaystyle=\mathbb{E}\left[\sum_{k=0}^{\infty}\left\langle
S\left(\pi\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}\right]$
$\displaystyle=\mathbb{E}\left[K^{\pi\gamma,\sigma}(s,t)\right].$

We conclude the proof. ∎

######  Remark 3.4.

If the random variable $\pi$ has a known probability density function, the
expectation in equation (3.8) can be calculated by numerical methods such as
Monte Carlo method or Gaussian quadrature procedure.

The corollary below gives two specialisations of this result to the cases
described earlier.

######  Corollary 3.5.

Let $\gamma,\sigma:[a,b]\rightarrow V$ be two continuous paths of bounded
variation.

(1) The $\phi$-signature kernel under $\phi(k)=\left(\frac{k}{2}\right)!$
satisfies

(3.9)
$K_{\phi}^{\gamma,\sigma}(s,t)=\mathbb{E}_{\pi}\left[K^{\pi^{1/2}\gamma,\sigma}(s,t)\right]=\mathbb{E}_{\pi}\left[K^{\gamma,\pi^{1/2}\sigma}(s,t)\right],$

where $\pi\sim\text{Exp}(1)$ is an exponentially distributed random variable
with intensity $1$.

(2) The $\phi$-signature kernel $K_{\phi}^{\gamma,\sigma}(s,t)$ under
$\phi(k)=\frac{\Gamma(m+1)\Gamma(k+1)}{\Gamma(k+m+1)}$ satisfies equation
(3.8) where $\pi\sim B(1,m)$ is a Beta-distributed random variable.

###### Proof.

For (1), we need to show that $\phi$ is all the moments of the random variable
$\pi^{1/2}$. Since $\pi\sim\text{Exp}(1)$, we have

$\mathbb{E}\left[\text{$\pi$}^{k/2}\right]=\int_{0}^{\infty}x^{k/2}e^{-x}dx=\Gamma\left(\frac{k}{2}+1\right)=\phi(k).$

The equation (3.9) then follows from Theorem 3.3. For (2), since the random
variable $\pi$ is Beta distributed, i.e. $\pi\sim\textrm{Beta}(1,m)$, then the
moments of $\pi$ are

$\mathbb{E}[\pi^{k}]=\frac{B(k+1,m)}{B(1,m)}=\frac{\Gamma(k+1)\Gamma(m+1)}{\text{$\Gamma$}(k+m+1)}=\phi(k).$

We conclude the proof. ∎

The motivation for the representation (3.8) is that we can design efficient
and accurate computational methods to compute the $\phi$-signature kernels. We
will give details on the Gaussian quadrature methods for the $\phi$-signature
kernel in Section 4 below.

###  3.3. General Signature Kernels by Fourier Series

We now extend the earlier discussion so that $\phi:\mathbb{Z}\rightarrow$
$\mathbb{C}$ is a complex-valued function. We consider the blinear form
defined by the two-sided summation

$\left\langle
a,b\right\rangle_{\phi}:=B_{\phi}\left(a,b\right):=\sum_{k=-\infty}^{\infty}\phi\left(k\right)\left\langle
a_{|k|},b_{|k|}\right\rangle_{|k|},$

and the corresponding function

$K_{\phi}^{\gamma,\sigma}\left(s,t\right):=\left\langle
S\left(\gamma\right)_{a,s},S\left(\sigma\right)_{a,t}\right\rangle_{\phi}.$

If the coefficients are the Fourier coefficients of some known periodic
function $f$ then the idea of the previous proposition can be applied to again
derive a representation of $K_{\phi}^{\gamma,\sigma}$. The following result
describes the needed conditions.

######  Proposition 3.6.

Suppose that $\gamma$ and $\sigma$ are continuous paths of bounded
$1$-variation. Let $\phi:\mathbb{Z}\rightarrow$ $\mathbb{C}$ be as above, and
write $\phi_{k}:=\phi\left(k\right).$ Assume that
$\left\\{\phi_{k}:k\in\mathbb{N}\right\\}$ are the Fourier coefficients of
some bounded integrable function $f:(-\pi,\pi)\rightarrow\mathbb{C},$ i.e.

$f=\sum_{k=-\infty}^{\infty}\phi_{k}e_{k},\text{ with
}e_{k}\left(x\right):=e^{ikx}.$

Then for all $\left(s,t\right)\in\left[a,b\right]\times\left[a,b\right]$ we
have

(3.10)
$K_{\phi}^{\gamma,\sigma}\left(s,t\right)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\bar{K}_{x}^{\gamma,\sigma}\left(s,t\right)f\left(x\right)dx-\phi_{0},$

where

$\bar{K}_{x}^{\gamma,\sigma}\left(s,t\right):=K^{\exp\left(-ix\right)\gamma,\sigma}\left(s,t\right)+K^{\exp\left(ix\right)\gamma,\sigma}\left(s,t\right).$

###### Proof.

Fixing $\left(s,t\right)$, we have for every $x\in\left(-\pi,\pi\right)$ that

$K^{\exp\left(\pm
ix\right)\gamma,\sigma}\left(s,t\right)=\sum_{k=0}^{\infty}e_{\pm
k}\left(x\right)\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}=:\sum_{k=0}^{\infty}e_{\pm
k}\left(x\right)c_{k}.$

The basic estimate $\left|c_{k}\right|\leq
L_{\gamma}^{k}L_{\sigma}^{k}/\left(k!\right)^{2}$ where $L_{\gamma}$ is the
length of the path $\gamma$ ensures that $\sum_{k=0}^{N}c_{k}e_{\pm
k}\left(\cdot\right)f\left(\cdot\right)$ converges uniformly to the series
$\sum_{k=0}^{\infty}c_{k}e_{\pm k}\left(\cdot\right)f\left(\cdot\right)$ and
hence

$\frac{1}{2\pi}\int_{-\pi}^{\pi}K^{\exp\left(\pm
ix\right)\gamma,\sigma}\left(s,t\right)f\left(x\right)dx=\frac{1}{2\pi}\sum_{k=0}^{\infty}c_{k}\int_{-\pi}^{\pi}e_{\pm
k}\left(x\right)f\left(x\right)dx=\sum_{k=0}^{\infty}c_{k}\phi_{\mp k}.$

It follows that

$\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}\left[K^{\exp\left(-ix\right)\gamma,\sigma}\left(s,t\right)+K^{\exp\left(ix\right)\gamma,\sigma}\left(s,t\right)\right]f\left(x\right)dx$
$\displaystyle=\sum_{k=-\infty}^{\infty}c_{|k|}\phi_{k}+c_{0}\phi_{0}=K_{\phi}^{\gamma,\sigma}\left(s,t\right)+\phi_{0},$

as required. ∎

######  Remark 3.7.

Note that
$\mathcal{R}K_{x}^{\gamma,\sigma}\left(s,t\right):=\operatorname{Re}K^{\exp\left(ix\right)\gamma,\sigma}\left(s,t\right)$
so that

$\mathcal{R}K_{x}^{\gamma,\sigma}\left(s,t\right)=\sum_{k=0}^{\infty}\cos
kx\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}.$

Together with
$\mathcal{I}K_{x}^{\gamma,\sigma}\left(s,t\right):=\operatorname{Im}K^{\exp\left(ix\right)\gamma,\sigma}\left(s,t\right)$
it solves the 2-dimensional PDE

$\frac{\partial^{2}}{\partial s\partial
t}\left(\begin{array}[]{c}\mathcal{R}K_{x}^{\gamma,\sigma}\left(s,t\right)\\\
\mathcal{I}K_{x}^{\gamma,\sigma}\left(s,t\right)\end{array}\right)=\left(\begin{array}[]{cc}\cos
x&-\sin x\\\ \sin x&\cos
x\end{array}\right)\left(\begin{array}[]{c}\mathcal{R}K_{x}^{\gamma,\sigma}\left(s,t\right)\\\
\mathcal{I}K_{x}^{\gamma,\sigma}\left(s,t\right)\end{array}\right)\left\langle\gamma_{s}^{\prime},\sigma_{t}^{\prime}\right\rangle.$

######  Corollary 3.8.

Special cases of the above result include:

  1. (1)

If $\phi_{k}=0$ for $k<0$ then

$K_{\phi}^{\gamma,\sigma}\left(s,t\right)=\frac{1}{2\pi}\int_{-\pi}^{\pi}K^{\exp\left(-ix\right)\gamma,\sigma}\left(s,t\right)f\left(x\right)dx.$

  2. (2)

(Real Fourier series) Suppose

$f=a_{0}+\sum_{k=1}^{\infty}a_{k}c_{k}+\sum_{k=1}^{\infty}b_{k}s_{k},\text{
where
}c_{k}\left(\cdot\right):=\cos\left(k\cdot\right),~{}s_{k}\left(\cdot\right):=\sin\left(k\cdot\right)$

with $\left\\{a_{k}\right\\}$ and $\left\\{b_{k}\right\\}$ real sequences. If

(3.11) $\left\langle
p,q\right\rangle_{\phi}:=\sum_{k=0}^{\infty}a_{k}\left\langle
p_{k},q_{k}\right\rangle_{k},$

then

$K_{\phi}^{\gamma,\sigma}\left(s,t\right)=\frac{1}{\pi}\int_{-\pi}^{\pi}\mathcal{R}K_{x}^{\gamma,\sigma}\left(s,t\right)f\left(x\right)dx-
a_{0}.$

In using this result the function $f$ should be chosen that the integral can
easily approximated numerically.

######  Example 3.9.

The following simple examples illustrate the scope of these ideas.

  1. (1)

The function $f\left(x\right)=x^{2}$ has the Fourier series
$f=\sum_{k=-\infty}^{\infty}\phi_{k}e_{k}$ on $\left[-\pi,\pi\right]$ where

$\phi_{k}=\frac{4\left(-1\right)^{k}}{k^{2}},\phi_{0}=\frac{\pi^{2}}{3},$

and we obtain the identity

$\sum_{k=1}^{\infty}\frac{4\left(-1\right)^{k}}{k^{2}}\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}=\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathcal{R}K_{x}^{\gamma,\sigma}\left(s,t\right)x^{2}dx-\frac{\pi^{2}}{3}.$

  2. (2)

The periodic function $f\left(x\right)=e^{\cos x}\cos\left(\sin x\right)$ has
Fourier series

$f\left(x\right)=\sum_{k=0}^{\infty}\frac{1}{k!}\cos\left(kx\right)$

and so

$\sum_{k=0}^{\infty}\frac{1}{k!}\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}=\frac{1}{\pi}\int_{-\pi}^{\pi}\mathcal{R}K_{x}^{\gamma,\sigma}\left(s,t\right)e^{\cos
x}\cos\left(\sin x\right)dx-1.$

  3. (3)

The Jacobi theta function is the $1$-periodic function

$\theta\left(z;\tau\right)=1+2\sum_{k=1}^{\infty}e^{i\pi\tau
k^{2}}\cos\left(2\pi kz\right),$

hence if we define
$f\left(x;u\right):=\theta\left(\frac{x}{2\pi};\frac{iu}{\pi}\right),$ then
$f\left(\cdot;u\right)=1+\sum_{k=1}^{\infty}e^{-uk^{2}}c_{k}$ and

$\sum_{k=0}^{\infty}e^{-uk^{2}}\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}=\frac{1}{\pi}\int_{-\pi}^{\pi}\mathcal{R}K_{x}^{\gamma,\sigma}\left(s,t\right)f\left(x;u\right)dx-1.$

###  3.4. General Signature Kernels by Integral Transforms

The main idea of the previous subsection was to look for a function $f$ with
Fourier series $\sum_{k\in\mathbb{Z}}\phi\left(k\right)e_{k}.$ If such a
function can be found, then we can calculate the bilinear form $B_{\phi}$
evaluated at a pair of signatures. The difficulty with this approach is that
such a function may not exist in some cases of interest, e.g.
$\phi\left(k\right)=k^{-1/2},$ $\phi\left(k\right)=k!$ etc. To simplify we
forego the two-sided summation, and re-define

$\left\langle
a,b\right\rangle_{\phi}:=B_{\phi}\left(a,b\right):=\sum_{k=0}^{\infty}\phi\left(k\right)\left\langle
a_{k},b_{k}\right\rangle_{k},$

where $\phi:\mathbb{R}\rightarrow\mathbb{C}$ is now defined on $\mathbb{R}$.
We assume that $\phi$ is the integral of a function $r:\mathbb{R\times
R\rightarrow}\mathbb{C}$ against a finite signed Borel measure $\mu$ on
$\mathbb{R}$ such that

(3.12) $\phi\left(u\right)=\int_{C}r\left(u,z\right)\mu\left(dz\right);\text{
where }r\left(u,z\right)=g\left(z\right)^{\alpha u}\text{ }\in\mathbb{C}\text{
for }\alpha\in\mathbb{R}.$

######  Example 3.10.

We will consider three principal examples:

  1. (1)

Fourier-Stieltjes transform: $C=\mathbb{R},\ g\left(z\right)=e^{-2\pi iz},\
\alpha=1$, i.e.
$\phi\left(u\right)=\hat{\mu}\left(u\right):=\int_{\mathbb{R}}e^{-2\pi
iuz}\mu\left(dz\right);$

  2. (2)

Laplace-Stieltjes transform: $C=\left(0,\infty\right),\
g\left(z\right)=e^{-z},\ \alpha=1$, i.e.
$\phi\left(u\right)=\tilde{\mu}\left(u\right):=$
$\int_{0}^{\infty}e^{-uz}\mu\left(dz\right);$

  3. (3)

Mellin-Stieltjes transform: $C=\left(0,\infty\right),\ g\left(z\right)=z,\
\alpha=1$, i.e.
$\phi\left(u\right)=\mu_{\text{Mel}}\left(u+1\right)=\int_{0}^{\infty}z^{u}\mu\left(dz\right)$,
$\operatorname{Re}u>-1.$

In the general case we can expect - under reasonable assumptions - that the
integral representation can be used to justify the calculation

$\displaystyle\left\langle
S\left(\gamma\right)_{a,s},S\left(\sigma\right)_{a,t}\right\rangle_{\phi}$
$\displaystyle=\sum_{k=0}^{\infty}\int_{C}g\left(z\right)^{\alpha
k}\mu\left(dz\right)\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}$
(3.13) $\displaystyle=\int_{C}\sum_{k=0}^{\infty}\left\langle
S\left(g\left(z\right)^{\alpha}\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}\mu\left(dz\right)$
$\displaystyle=\int_{C}K^{g\left(z\right)^{\alpha}\gamma,\sigma}\left(s,t\right)\mu\left(dz\right).$

again allowing us to reduce the calculation of the the bilinear form to a
weighted integral over PDE solutions. On this occasion integration is w.r.t.
the measure $\mu$ and the rescaling is determined by the form of the kernel
function $r$ in the integral transform relating $\mu$ and $\phi$.

######  Theorem 3.11.

Let $\mu$ be a finite signed Borel measure $\mu$ on $\mathbb{R}$. Suppose that
$\phi:\mathbb{R}\rightarrow\mathbb{C}$ is such that

$\phi\left(k\right)=\int_{C}r\left(k,z\right)\mu\left(dz\right)\in\mathbb{C}\,\text{,
for all }k\in\mathbb{N}\cup\\{0\\}$

where $r\left(u,\cdot\right)$ is assumed to have the form $r\left(u,z\right)=$
$g\left(z\right)^{\alpha u}$ $\in\mathbb{C}$ for $\alpha\in\mathbb{R}$ and
some function $g:\mathbb{C}\rightarrow\mathbb{C}.$ Let

$\gamma:\left[a,b\right]\rightarrow V\text{ and
}\sigma:\left[a,b\right]\rightarrow V$

be continuous paths of bounded $1$-variation with signatures
$S\left(\gamma\right)$ and $S\left(\sigma\right)$ respectively. For every
$\left(s,t\right)\in\left[a,b\right]\times\left[a,b\right]$ and
$k\in\mathbb{N\cup}\left\\{0\right\\}$ define

$a_{k}\left(s,t\right):=\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}.$

Assume for every $\left(s,t\right)\in\left[a,b\right]\times\left[a,b\right]$
that

  1. (1)

the integral
$\int_{C}\left|r\left(k,z\right)\right|\left|\mu\left(dz\right)\right|<\infty,$
and

  2. (2)

the series
$\sum_{k}a_{k}\left(s,t\right)\int_{C}\left|r\left(k,z\right)\right|\left|\mu\left(dz\right)\right|$
converges absolutely,

then

(3.14) $\left\langle
S\left(\gamma\right)_{a,s},S\left(\sigma\right)_{a,t}\right\rangle_{\phi}=\int_{C}K^{g\left(z\right)^{\alpha}\gamma,\sigma}\left(s,t\right)\mu\left(dz\right).$

######  Remark 3.12.

Sufficient for item 2 is that
$\sum_{k}\left(k!\right)^{-2}\int_{C}\left|h_{k}\left(z;s,t\right)\right|\left|dz\right|$
converges.

###### Proof.

Assumptions 1 and 2 above ensure that Fubini’s Theorem can be applied to give

$\sum_{k=0}^{\infty}a_{k}\left(s,t\right)\int_{C}r\left(k,z\right)\mu\left(dz\right)=\int_{C}\sum_{k=0}^{\infty}a_{k}\left(s,t\right)r\left(k,z\right)\mu\left(dz\right),$

which can be seen to be the same as (3.14) using the fact
$r\left(u,z\right)\equiv$ $g\left(z\right)^{\alpha u}.$ ∎

######  Corollary 3.13.

For each of the three integral transforms in Example 3.10 satisfying
assumption 1 and 2 in the above theorem, we have (3.14).

In a similar way we have the following results once again.

######  Corollary 3.14.

Let $\pi$ be a random variable with finite moments of all orders and

$\phi\left(k\right)=\mathbb{E}\left[\pi^{k}\right]\ \text{and}\
\psi\left(k\right)=\mathbb{E}\left[|\pi|^{k}\right]\text{ for all
}k\in\mathbb{N}\cup\left\\{0\right\\}$

such that $\psi$ satisfies Condition 1. Then

$K_{\phi}^{\gamma,\sigma}(s,t)=\mathbb{E}_{\pi}\left[K^{\pi\gamma,\sigma}(s,t)\right]=\mathbb{E}_{\pi}\left[K^{\gamma,\pi\sigma}(s,t)\right].$

###### Proof.

Let $F$ be the distribution function of $\pi$. Apply Theorem 3.11 with
$\mu=dF$ and $r\left(u,z\right)=z^{u}.$ ∎

######  Example 3.15.

These examples illustrate these results

  1. (1)

For any $\beta>-1,$ the function
$\phi\left(u\right)=\Gamma\left(u+\beta+1\right)=\int_{0}^{\infty}x^{u}x^{\beta}e^{-x}dx$
is the Mellin transform of $x^{\beta}e^{-x}$. Therefore, we have

$\sum_{k=0}^{\infty}\Gamma\left(k+\beta+1\right)\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}=\int_{0}^{\infty}K^{x\gamma,\sigma}\left(s,t\right)x^{\beta}e^{-x}dx.$

  2. (2)

Suppose $\pi$ is a random variable, the expectation can be computed in the
following cases:

    1. (a)

if $\pi$ is uniformly distributed on $\left[0,1\right],$ then it equals

$\sum_{k=0}^{\infty}\frac{1}{k+1}\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}=\int_{0}^{1}K^{x\gamma,\sigma}\left(s,t\right)dx;$

    2. (b)

if $\pi$ has the Arcsine$\left(-1,1\right)$-distribution, i.e.
$F_{\pi}\left(x\right)=\frac{2}{\pi}\arcsin\left(\sqrt{\frac{1+x}{2}}\right)$
, then:

(3.15) $\sum_{k=0}^{\infty}\prod_{r=0}^{2k-1}\frac{2r+1}{2r+2}\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}=\frac{1}{\pi}\int_{-1}^{1}\frac{K^{x\gamma,\sigma}\left(s,t\right)}{\sqrt{1-x^{2}}}dx;$

    3. (c)

if $\pi$ has the Beta$\left(\alpha,\beta\right)$-distribution, then:

$\sum_{k=0}^{\infty}\prod_{r=0}^{k-1}\frac{\alpha+\beta}{\alpha+\beta+r}\left\langle
S\left(\gamma\right)_{a,s}^{k},S\left(\sigma\right)_{a,t}^{k}\right\rangle_{k}=\frac{1}{B(\alpha,\beta)}\int_{0}^{1}K^{x\gamma,\sigma}(s,t)x^{\alpha-1}\left(1-x\right)^{\beta-1}dx.$

##  4\. Computing General Signature Kernels

The usefulness of the formulae in the last section depend on being able to
numerically approximate integrals such as

$\int_{a}^{b}f\left(x\right)w\left(x\right)dx$

where $[a,b]\subseteq\mathbb{R}$, $w\in L^{1}\left(\left(a,b\right)\right)$ is
a weight function, which for the moment we assume to be positive. In the
examples considered the function $f$ to be integrated will be a scaling of the
signature kernel PDE, typically we will have

$f\left(x\right)=K^{x\gamma,\sigma}(s,t).$

The classical approach to such approximations is to use a Gaussian Quadrature
Rule, see e.g. [32]

For a general weight function, suppose that
$\mathcal{P=}\left\\{p_{n}:n\in\mathbb{N}\cup\left\\{0\right\\}\right\\}$ is a
system of orthogonal polynomials w.r.t. the weight function $w\,$ over
$\left(a,b\right)$; that is $\deg\left(p_{n}\right)=n$ and $\left\langle
p_{n},p_{m}\right\rangle_{w}=\int_{a}^{b}p_{m}p_{n}wdx=0$ for $n\neq m.$ Then
the quadrature points $x_{k}$, $k=0,1,...,n$ are the zeros of the polynomial
$p_{n+1}$, the corresponding quadrature weights are

$w_{k}:=\int_{a}^{b}w\left(x\right)\prod_{i=0,i\neq
k}^{n}\left(\frac{x-x_{i}}{x_{k}-x_{i}}\right)^{2}dx$

and the quadrature rule is the approximation

$\int_{a}^{b}f\left(x\right)w\left(x\right)dx\approx\sum_{k=0}^{n}w_{k}f\left(x_{k}\right).$

The approximation is exact if $f$ is a polynomial with $\deg\left(f\right)\leq
2n+1.$ If $f$ is assumed to be $C^{2n+2},$ then the error in the quadrature
rule can be approximated by the basic estimate [32]

(4.1)
$\left|\int_{a}^{b}f\left(x\right)w\left(x\right)dx-\sum_{k=0}^{n}w_{k}f\left(x_{k}\right)\right|\leq\frac{f^{\left(2n+2\right)}\left(\xi\right)}{\left(2n+2\right)!}\int_{a}^{b}w\left(x\right)\pi_{n+1}\left(x\right)^{2}dx,$

where $\xi\in\left(a,b\right)$ and

$\pi_{n+1}\left(x\right)=\prod_{i=0}^{n}\left(x-x_{i}\right)$

is the monic poynomial obtained by dividing $p_{n+1}$ by its leading
coefficient. In view of the bound (4.1) we have the following

######  Lemma 4.1.

Define $f\left(x\right):=K^{x\gamma,\sigma}(s,t)$ for $x\in\mathbb{R}.$ Then
$f$ is infinitely differentiable and, for every $k\in\mathbb{N}$ , its $k$th
derivative is given by

(4.2)
$f^{\left(k\right)}\left(x\right)=\sum_{l=0}^{\infty}x^{l}\frac{\left(l+k\right)!}{l!}\left\langle
S\left(\gamma\right)_{a,s}^{l+k},S\left(\sigma\right)_{a,t}^{l+k}\right\rangle_{l+k}.$

In particular, we have the estimate

(4.3)
$\left|f^{\left(k\right)}\left(x\right)\right|\leq\frac{L_{s}\left(\gamma\right)^{k/2}L_{t}\left(\sigma\right)^{k/2}}{|x|^{k/2}}I_{k}\left(2\sqrt{|x|L_{s}\left(\gamma\right)L_{t}\left(\sigma\right)}\right),$

where $L_{s}(\gamma)$ is the length of the path segment $\gamma|_{[a,s]}$ and
$I_{k}$ is the modified Bessel function of the first kind of order $k.$

###### Proof.

Differentiablity is a simple argument on term-by-term differentiation of power
series. Applying this argument $k$ times results in the formula (4.2). The
bound (4.3) can be obtained by the elementary estimate

$\displaystyle\left|f^{\left(k\right)}\left(x\right)\right|$
$\displaystyle\leq\sum_{l=0}^{\infty}|x|^{l}\frac{\left(l+k\right)!}{l!}\frac{L_{s}\left(\gamma\right)^{l+k}L_{t}\left(\sigma\right)^{l+k}}{\left(l+k\right)!^{2}}$
$\displaystyle=\frac{L_{s}\left(\gamma\right)^{k/2}L_{t}\left(\sigma\right)^{k/2}}{|x|^{k/2}}I_{k}\left(2\sqrt{|x|L_{s}\left(\gamma\right)L_{t}\left(\sigma\right)}\right).$

∎

For any $x\in\mathbb{R},$ $k\in\mathbb{N}$ it is easy to derive from (4.2) the
crude estimate

$\left|f^{\left(k\right)}\left(x\right)\right|\leq\frac{L_{s}\left(\gamma\right)^{k}L_{t}\left(\sigma\right)^{k}}{k!}\exp\left(|x|L_{s}\left(\gamma\right)L_{t}\left(\sigma\right)\right),$

which could be refined e.g. by considering estimate on ratios of Bessel
functions $I_{k+1}/I_{k}.$ Putting things together we obtain.

######  Proposition 4.2.

Let $\mathcal{P=}\left\\{p_{n}:n\in\mathbb{N}\cup\left\\{0\right\\}\right\\}$
be a system of orthogonal polynomials with respect to a continuous positive
weight function $w\in L^{1}\left(a,b\right).$ For every $n$ the error in the
associated quadrature is bounded above by

$\displaystyle\quad\left|\int_{a}^{b}K^{x\gamma,\sigma}(s,t)w\left(x\right)dx-\sum_{k=0}^{n}w_{k}K^{x_{k}\gamma,\sigma}(s,t)\right|$
$\displaystyle\leq\frac{L_{s}\left(\gamma\right)^{2n+2}L_{t}\left(\sigma\right)^{2n+2}\exp\left(|\xi|L_{s}\left(\gamma\right)L_{t}\left(\sigma\right)\right)}{[\left(2n+2\right)!]^{2}}\int_{a}^{b}w\left(x\right)\pi_{n+1}\left(x\right)^{2}dx.$

######  Example 4.3.

Let $\left(a,b\right)=\left(-1,1\right),$
$w\left(x\right)=\frac{1}{\pi}\frac{1}{\sqrt{1-x^{2}}}$ as in the earlier
example (3.15). Then $\mathcal{P}$ can be the family of Chebyshev polynomials
of the first kind $p_{n}=T_{n}$ in which case (see [1])

$\int_{a}^{b}w\left(x\right)\pi_{n+1}\left(x\right)^{2}dx=\frac{1}{2^{2n+1}}.$

Therefore if $\gamma$ and $\sigma$ have lengths at most $L$ the degree $n+1$
quadrature rule results in an error at most

$R_{n}=\frac{L^{4n+4}\exp\left(L^{2}\right)}{2^{2n+1}[\left(2n+2\right)!]^{2}}.$

To give some idea of the number of points needed (and hence the number of PDEs
solutions needed), if $L=10$ then $R_{25}=e^{-8.6017}$, $R_{30}=e^{-50.492},$
whereas if $L=100$ then $R_{1050}=e^{-49.497}$. The ratio

$\frac{R_{n+1}}{R_{n}}=\frac{L^{4}}{4(2n+4)^{2}(2n+3)^{2}},$

articulates the trade off between the length $L$ and the number of points $n$.

######  Example 4.4.

The $\phi$-signature kernel $K_{\phi}^{\gamma,\sigma}(s,t)$ for
$\phi(k)=\left(\frac{k}{2}\right)!$ is studied in Corollary 3.5. In this case
the random variable $\pi$ is exponentially distributed, hence $\pi^{1/2}$ is
Rayleigh distributed with density $w(x)=2xe^{-x^{2}},\ x>0$. We have

(4.4)
$K_{\phi}^{\gamma,\sigma}(s,t)=\mathbb{E}\left[K^{\pi^{1/2}\gamma,\sigma}(s,t)\right]=\int_{0}^{\infty}2K^{x\gamma,\sigma}(s,t)xe^{-x^{2}}dx.$

Let $f(x)=K^{x\gamma,\sigma}(s,t)$, then

$K_{\phi}^{\gamma,\sigma}(s,t)=2\int_{0}^{\infty}f(x)xe^{-x^{2}}dx$

which can be numerically calculated by the classical Gaussian quadrature
formula (see e.g. [28, 30]),

$\int_{0}^{\infty}f(x)xe^{-x^{2}}dx\approx\sum_{k=0}^{n}w_{k}f(x_{k}).$

The abscissae $x_{k},\ k=0,1,\cdots,n$ are the roots of a $(n+1)$-th degree
polynomial $p_{n+1}(x)$ and $w_{k}$ are the weights of quadrature. Explicit
values are given in [28, 30].

##  5\. Expected General Signature Kernels

We develop our earlier discussion to consider how $\phi$-signature kernels can
be combined with the notion of expected signatures to compare the laws of two
stochastic processes. In the examples we study one of the measures will be
Wiener’s measure, which we denote by $\mathcal{W}$ and the other will be
denote by $\mu$. The measure $\mu$ will typically discrete and supported on
bounded variation paths, Our aim will be to compute

$K_{\phi}^{\mathcal{W},\mu}\left(s,t\right)=\left\langle\mathbb{E}_{X\sim\mathcal{W}}\left[S\left(X\right)_{0,1}\right],\mathbb{E}_{X\sim\mu}\left[S\left(X\right)_{0,1}\right]\right\rangle_{\phi},$

where $S\left(X\right)$ denotes the Stratonovich signature of $X$. We will
sometimes write $\mathbb{E}\left[S\left(\circ B\right)_{0,s}\right]$, for a
Brownian motion $B$, in place of
$\mathbb{E}_{X\sim\mathcal{W}}\left[S\left(X\right)_{0,1}\right]$ to emphasise
the fact that the signature is constructed via Stratonvich calculus.

As an initial step, we assume that $\gamma$ is a fixed (deterministic)
continuous path of bounded variation. We look to obtain formula for the
$\phi$-signature kernel of the expected Stratonovich signature of Brownian
motion and $\gamma$, i.e.

$K_{\phi}^{\mathcal{W},\gamma}\left(s,t\right):=\left\langle\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right],S\left(\gamma\right)_{0,t}\right\rangle_{\phi}$

A key idea to doing this will be to use notion of the hyperbolic development
of $\gamma$ which has been used in earlier study of the signature and, in this
context, was initiated by [16]. We summarise the essential background in the
section below.

###  5.1. Hyperbolic Development

We gather the basic notation and results. Readers seeking further details can
consult the references [2, 16, 26]. We let $\mathbb{H}^{d}$ denote
$d$-dimensional hyperbolic space realised as the hyperboloid
$\\{x\in\mathbb{R}^{d+1}:x\ast x=-1,\ x_{d+1}>0\\}$ endowed with the Minkowski
product

$x\ast y=\sum_{i=1}^{d}x_{i}y_{i}-x_{d+1}y_{d+1}\text{, for
}x=(x_{1},\cdots,x_{d},x_{d+1})\in\mathbb{R}^{d+1}.$

It is well known that this defines a Riemannian metric when restricted to the
tangent bundle of $\mathbb{H}^{d}$. We let $\operatorname{d_{\mathbb{H}^{d}}}$
denote the associated Riemannian distance function and recall that

(5.1) $\cosh\operatorname{d_{\mathbb{H}^{d}}}(x,y)=-x\ast y,$

see e.g. [4]. Define the linear map
$F:\mathbb{C}^{d}\rightarrow\mathcal{M}_{d+1}\left(\mathbb{C}\right)$ into the
space of $d+1$ by $d+1$ matrices over $\mathbb{C}$ by

(5.2) $F:x\rightarrow\begin{pmatrix}0&x\\\ x^{T}&0\end{pmatrix}.$

Then if $V$ is a real inner product space of dimension $d$ and
$\gamma:[a,b]\rightarrow V$ is continuous path of bounded variation then, by
fixing an orthonormal basis of $V,$ and writing $\gamma$ in this basis as
$\gamma=(\gamma_{t_{1}},...,\gamma_{t_{d}})$ we can solve the linear
differential equation

(5.3) $d\Gamma_{s,t}(u)=F(d\gamma(u))\Gamma_{s,t}(u),\
u\in[s,t]\subset[a,b],\text{with }\Gamma_{s,t}(s)=I=I_{d+1}$

uniquely. In the case the map
$\gamma|_{[s,t]}\mapsto\Gamma_{s,t}\left(\cdot\right)$ takes a path segment in
$V$ into one in the isometry group of $\mathbb{H}^{d}.$ The resulting
$\Gamma_{s,t}\left(\cdot\right)~{}$ is called the Cartan Development of the
path segment $\gamma|_{[s,t]}.$ It satisfies the multiplicative property

(5.4) $\Gamma_{u,t}(t)\Gamma_{s,u}(u)=\Gamma_{s,t}(t),\ s\leq u\leq t.$

To simplify things we write $\Gamma(t):=\Gamma^{\gamma}(t):=\Gamma_{a,b}(t)$
for $t\in[a,b]$. It is elementary to represent $\Gamma$ as the convergent
series

(5.5)
$\Gamma(t)=I+\sum_{n=1}^{\infty}\int_{a<t_{1}<\cdots<t_{n}<t}F(d\gamma({t_{1}}))\cdots
F(d\gamma({t_{n}})).$

Then letting $o=(0,\cdots,0,1)^{T}\in\mathbb{H}^{d},$we define
$\sigma(t):=\Gamma(t)o$ to be the hyperbolic development of the path $\gamma$
onto $\mathbb{H}^{d}$, and we write $\sigma_{\gamma}$ to emphasise the
dependence on $\gamma.$

A global coordinate chart for $\mathbb{H}^{d}$ is determined by
$\mathbb{H}^{d}\ni m\mapsto\left(\eta,\rho\right)\in\mathbb{S}^{d-1}\times$
$\mathbb{R}_{+}$ where $\left(\eta\sinh\rho,\cosh\rho\right)=m.$ Using these
coordinates, we define

$\eta\left(t\right)=\eta_{\gamma}(t)=\eta\left(\sigma_{\gamma}\left(t\right)\right)\in\mathbb{S}^{d-1}\text{
and
}\rho\left(t\right)=\rho_{\gamma}\left(t\right)=\rho\left(\sigma_{\gamma}\left(t\right)\right)\in\mathbb{R}_{+}.$

The following identity follows from (5.5) and (5.1):

(5.6)
$\cosh\rho_{\gamma}(t)=\Gamma_{d+1,d+1}(t)=1+\sum_{n=1}^{\infty}\int_{a<t_{1}<\cdots<t_{2n}<t}\langle
d\gamma({t_{1}}),d\gamma({t_{2}})\rangle\cdots\langle
d\gamma({t_{2n-1}}),d\gamma({t_{2n}})\rangle,$

where
$\Gamma\left(t\right)=\left(\Gamma_{ij}\left(t\right)\right)_{i,j=1,\cdots,d+1}$.
We will need to broaden this discussion to consider the development of paths
after complex rescaling. To this end, if $\gamma$ is as above and
$z\in\mathbb{C}$ then we let $z\gamma$ denote the path in $V^{\mathbb{C}}$,
the complexification of $V$. We will be interested in the relationship between
the solution to (5.3), when $\gamma$ is replaced by $z\gamma$, and the series
(5.6). The following lemma identifies the structure we need.

######  Lemma 5.1.

Let $\gamma:[a,b]\rightarrow V$ be a continuous path of bounded variation. For
$z\in\mathbb{C}$ let $z\gamma:\left[a,b\right]\rightarrow V^{\mathbb{C}}$ be
the rescaling of $\gamma$ by $z\in\mathbb{C}$. Given an orthonormal basis of
$V$, write $\gamma_{t}=$
$\left(\gamma_{t}^{1},...,\gamma_{t}^{d}\right)\in\mathbb{R}^{d}$ and
$z\gamma\left(t\right):=\left(z\gamma_{t}^{1},...,z\gamma_{t}^{d}\right)\in\mathbb{C}^{d}$
in terms of this basis. Then

(5.7) $d\Gamma^{z\gamma}(u)=F(d\left(z\gamma\right)(u))\Gamma^{z\gamma}(u),\
u\in[a,b],\text{with }\Gamma^{z\gamma}(s)=I_{d+1}$

has a unique solution in $\mathcal{M}_{d+1}\left(\mathbb{C}\right)$ and
furthermore the entry

(5.8)
$\Gamma_{d+1,d+1}^{z\gamma}(t)=1+\sum_{n=1}^{\infty}z^{2n}\int_{0<t_{1}<...<t_{2n}<t}\left\langle
d\gamma_{t_{1}},d\gamma_{t_{2}}\right\rangle...\left\langle
d\gamma_{t_{2n-1}},d\gamma_{t_{2n}}\right\rangle.$

If $\gamma$ is a piecewise linear path defined by the concatenation

$\gamma_{v_{1}}\ast\gamma_{v_{2}}....\ast\gamma_{v_{n}}:\left[a,b\right]\rightarrow
V,$

i.e. $\gamma$ is such that
$\gamma_{v_{i}}^{\prime}\left(t\right)=v_{i}\in\mathbb{R}^{d}$ for
$t\in(t_{i-1},t_{i})$. Then the solution to (5.7) is given explicitly by the
matrix product

(5.9)
$\Gamma^{z\gamma}\left(b\right)=A\left(v_{n},\Delta_{n},z\right)A\left(v_{n-1},\Delta_{n-1},z\right)\cdots
A\left(v_{1},\Delta_{1},z\right),$

where $\Delta_{i}=t_{t}-t_{i-1}$ and

(5.10)
$A\left(v,\Delta,z\right):=I_{d+1}+\sinh\left(z\left|v\right|\Delta\right)M+\left(\cosh\left(z\left|v\right|\Delta\right)-1\right)M^{2}$

in which

$M=\begin{pmatrix}0&\tilde{v}\\\
\tilde{v}^{T}&0\end{pmatrix}\in\mathcal{M}_{d+1}\left(\mathbb{R}\right)\text{
with }\tilde{v}=\frac{v}{\left|v\right|}.$

###### Proof.

Since the ODE (5.7) is linear, there is a unique solution
$\Gamma^{z\gamma}(t)$ which can be represented by equation (5.5) by replacing
$\gamma$ with $z\gamma$. Then equation (5.8) can be obtained by taking the
last entry of this equation.

To obtain the explicit solution in the case where $\gamma$ is piecewise linear
path, we first assume $\gamma^{\prime}=v$ on $[s,t]$. Then by using the
observation that $M^{3}=M$ together with equation (5.5), we have

$\displaystyle\Gamma_{s,t}^{z\gamma}(t)$
$\displaystyle=I+\sum_{n=1}^{\infty}\frac{(z|v|)^{2n-1}(t-s)^{2n-1}}{(2n-1)!}M+\sum_{n=1}^{\infty}\frac{(z|v|)^{2n}(t-s)^{2n}}{(2n)!}M^{2}$
$\displaystyle=I+\sinh\left(z\left|v\right|(t-s)\right)M+\left(\cosh\left(z\left|v\right|(t-s)\right)-1\right)M^{2}.$

In the general case, the multiplicative property (5.4) together with simple
induction argument implies that the solution has the form (5.9). ∎

###  5.2. Signature Kernels and Hyperbolic Development

We begin this subsection by giving a closed form of the $\phi$-signature
kernel $K_{\phi}^{\mathcal{W},\mu}\left(s,t\right)$ for the special case
$\phi(k)=\left(\frac{k}{2}\right)!$ based on the theory presented above.

######  Theorem 5.2.

(Formula for $\left\langle\mathbb{E}\left[S\left(\circ
B\right)\right],S\left(\gamma\right)\right\rangle_{\phi}$) Let
$\phi:\mathbb{N}\cup\left\\{0\right\\}\rightarrow\mathbb{R}_{+}$ be defined by
$\phi(k)=\left(\frac{k}{2}\right)!$ for $k\in\mathbb{N}\cup\\{0\\}$. Suppose
that $B$ is a $d$-dimensional Brownian motion, then the expected Stratonovich
signature, $\mathbb{E}\left[S\left(\circ B\right)_{0,s}\right]$, belongs to
$T_{\phi}\left(V\right)$ for any $0\leq s<\infty.$ Furthermore if
$\gamma:[0,1]\to V$ is any continuous path of bounded variation it holds that

(5.11)
$K_{\phi}^{\mathcal{W},\gamma}\left(s,t\right):=\left\langle\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right],S\left(\gamma\right)_{0,t}\right\rangle_{\phi}=\cosh\left(\rho_{\sqrt{s/2}\gamma}\left(t\right)\right).$

In this notation
$\rho_{\lambda}\left(t\right):=\operatorname{d_{\mathbb{H}^{d}}}\left(o,\sigma_{\lambda\gamma}\left(t\right)\right)$
is the distance between the hyperbolic development $\sigma_{\lambda\gamma}(t)$
of the path $\lambda\gamma\left(\cdot\right)$ from $T_{o}\mathbb{H}^{d}$ onto
the $d$-dimensional hyperbolic space $\mathbb{H}^{d}$ started at the base
point $o\in\mathbb{H}^{d}$, and
$\operatorname{d_{\mathbb{H}^{d}}}:\mathbb{H}^{d}\times\mathbb{H}^{d}\rightarrow[0,\infty)$
is the Riemannian distance on $\mathbb{H}^{d}$.

###### Proof.

For the first assertion recall that (see e.g. Proposition 4.10. in [25])

$\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right]=\exp\left(\frac{s}{2}\sum_{i=1}^{d}e_{i}^{2}\right)=\sum_{k=0}^{\infty}\frac{s^{k}}{2^{k}k!}\sum_{i_{1},...,i_{k}=1}^{d}e_{i_{1}}^{2}...e_{i_{k}}^{2}$

so that

$\left|\left|\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right]\right|\right|_{\phi}^{2}=\sum_{k=0}^{\infty}k!\frac{s^{2k}d^{k}}{2^{2k}(k!)^{2}}=e^{s^{2}d/4}<\infty.$

For the second assertion we have that

$\left\langle\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right],S\left(\gamma\right)_{0,t}\right\rangle_{\phi}=\sum_{k=0}^{\infty}\frac{s^{k}}{2^{k}}\int_{0<t_{1}<...<t_{2k}<t}\left\langle
d\gamma_{t_{1}},d\gamma_{t_{2}}\right\rangle...\left\langle
d\gamma_{t_{2k-1}},d\gamma_{t_{2k}}\right\rangle.$

The right hand side of this expression equals that of (5.11); see formula
(5.6). ∎

In the following, we give some remarks on the computation of this basic
signature kernel based on the above theorem.

######  Remark 5.3.

(1) In contrast to the earlier case of two paths, we need only solve an ODE to
calculate $\left\langle\mathbb{E}\left[S\left(\circ
B\right)\right],S\left(\gamma\right)\right\rangle_{\phi}$ and not a PDE. (2)
For general $\gamma$, the ODE is known, and is determined by the linear vector
fields in equation (5.3). Any ODE solver such as Runge-Kutta could in
principle be used to obtain numerical solutions. (3) For piecewise linear
case, the exact solution is given in equation (5.9) as a product of matrices.

###  5.3. The Original Kernel for Expected Signatures

Theorem 5.2 gives a closed form expression for the $\phi$-signature kernel of
Stratonovich expected signature of Brownian motion and the signature of a
bounded variation continuous path where $\phi(k)=\left(\frac{k}{2}\right)!$.
As previously we will be interested in related formulae for different
signature kernels. We can obtain these formulae by using an extension of the
ideas developed earlier in the paper. In the case of the original signature
kernel (i.e. $\phi\equiv 1$), we can make use of the classical integral
representation of the reciprocal gamma function which for integers has the
form:

(5.12) $\frac{1}{k!}=\frac{1}{2\pi
i}\oint_{C}z^{-(k+1)}e^{z}dz=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-ik\theta}e^{e^{i\theta}}d\theta$

where $\oint_{C}$ denotes the contour integral around the unit circle
traversed once anticlockwise. This is an instance of the more general formula

(5.13) $\frac{1}{\Gamma(p)}=\frac{1}{2\pi i}\oint_{H}z^{-p}e^{z}dz,$

where $H$ is Hankel contour which winds from $-\infty-0i$ in the lower half-
plane, anticlockwise around 0, and then back to $-\infty+0i$ in the upper
half-plane, while respecting the branch cut of the integrand along the
negative real axis. The advantage of using these integral representation is
twofold. First, the integrand has exponential dependence on $k$ making it
suitable to employ the techniques developed earlier in the paper. Second the
underlying numerical integration theory is well developed and the convergence
rates for optimised quadrature formulae are exceedingly fast. We give some
examples below but refer the reader to the reference [34] for further details.
We have the following theorem.

######  Theorem 5.4.

Let $\phi\equiv 1$. Suppose $B$ is a $d$-dimensional Brownian motion, then the
expected Stratonovich signature, $\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right]$, belongs to $T_{\phi}\left(V\right)$ for any $0\leq
s<\infty$ and

(5.14) $\left|\left|\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right]\right|\right|_{\phi}^{2}=\frac{1}{2\pi
i}\oint_{C}z^{-1}e^{z+s^{2}d/(4z)}dz$

where the contour $C$ is the unit circle in $\mathbb{C}$ traversed
anticlockwise. Furthermore if $\gamma$ is any continuous path of bounded
variation it holds that

(5.15)
$K_{\phi}^{\mathcal{W},\gamma}\left(s,t\right):=\left\langle\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right],S\left(\gamma\right)_{0,t}\right\rangle_{\phi}=\frac{1}{2\pi
i}\oint_{C}z^{-1}e^{z}\Gamma_{d+1,d+1}^{c_{s}(z)\gamma}(t)dz$

where $c_{s}(z)=\sqrt{s/2z}\in\mathbb{C}$ and
$\Gamma_{d+1,d+1}^{c_{s}(z)\gamma}(t)$ is defined by the series (5.8), i.e.
the last entry of the solution to ODE (5.7).

###### Proof.

Using the definition of the original signature kernel and the dominated
convergence theorem to interchange the order of $\sum$ and $\oint_{C}$ we have

$\displaystyle\left\langle\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right],S\left(\gamma\right)_{0,t}\right\rangle_{\phi}$
$\displaystyle=\sum_{k=0}^{\infty}\frac{1}{k!}\frac{s^{k}}{2^{k}}\int_{0<t_{1}<...<t_{2k}<t}\left\langle
d\gamma_{t_{1}},d\gamma_{t_{2}}\right\rangle...\left\langle
d\gamma_{t_{2k-1}},d\gamma_{t_{2k}}\right\rangle$ $\displaystyle=\frac{1}{2\pi
i}\oint_{C}z^{-1}e^{z}\left(\sum_{k=0}^{\infty}z^{-k}\frac{s^{k}}{2^{k}}\int_{0<t_{1}<...<t_{2k}<t}\left\langle
d\gamma_{t_{1}},d\gamma_{t_{2}}\right\rangle...\left\langle
d\gamma_{t_{2k-1}},d\gamma_{t_{2k}}\right\rangle\right)dz.$

If $c_{s}(z)=\sqrt{s/2z}$ then by equation (5.8), we know that

$\sum_{k=0}^{\infty}z^{-k}\frac{s^{k}}{2^{k}}\int_{0<t_{1}<...<t_{2k}<t}\left\langle
d\gamma_{t_{1}},d\gamma_{t_{2}}\right\rangle...\left\langle
d\gamma_{t_{2k-1}},d\gamma_{t_{2k}}\right\rangle=\Gamma_{d+1,d+1}^{c_{s}(z)\gamma}(t),$

which is the last entry of the solution $\Gamma^{c_{s,z}\gamma}(t)$ to ODE
(5.7). The argument for the squared norm of Brownian motion, follows a similar
pattern and yields

$\left|\left|\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right]\right|\right|_{\phi}^{2}=\frac{1}{2\pi
i}\oint_{C}z^{-1}e^{z}\left(\sum_{k=0}^{\infty}z^{-k}\frac{s^{2k}d^{k}}{2^{2k}k!}\right)dz=\frac{1}{2\pi
i}\oint_{C}z^{-1}e^{z}e^{s^{2}d/(4z)}dz.$

∎

#### Computation of the contour integrals

The implementation of the formula above demands an efficient way to
approximate contour integrals of the form

(5.16) $I=\frac{1}{2\pi
i}\oint_{C}e^{z}f(z)dz=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{e^{i\theta}}f(e^{i\theta})e^{i\theta}d\theta.$

A natural approach is to apply a trapezoidal rule based on $N$ equally spaced
points on the unit circle, i.e. to approximate $I$ using

(5.17) $I_{N}=\frac{1}{N}\sum_{k=1}^{N}e^{z_{k}}f(z_{k})z_{k}$

where $z_{k}=e^{2k\pi i/N}$. Several other methods have been proposed in
Trefethen, Weideman and Schmelzer (2006) [34] for the efficient approximation
of the Hankel-type contour integrals of the form

$I=\frac{1}{2\pi i}\oint_{H}e^{z}f(z)dz.$

The idea is to seek an optimal selection of contour according to the number of
points in the quadrature formula. Letting $\varphi(\theta)$ be an analytic
function that maps the real line $\mathbb{R}$ onto the contour $H$. Then the
approach is to approximate

$I=\frac{1}{2\pi
i}\int_{-\infty}^{+\infty}e^{\varphi(\theta)}f(\varphi(\theta))\varphi^{\prime}(\theta)d\theta$

by

(5.18)
$I_{N}=-iN^{-1}\sum_{k=1}^{N}e^{z_{k}}f(z_{k})w_{k}=-\sum_{k=1}^{N}c_{k}f(z_{k})$

on the finite interval $[-\pi,\pi]$ with $N$ points which are regularly spaced
on the interval and $z_{k}=\varphi(\theta_{k})$,
$w_{k}=\varphi^{\prime}(\theta_{k})$ and $c_{k}=iN^{-1}e^{z_{k}}w_{k}$. The
convergence rates for these optimised quadrature formulae are very fast, of
order $O(3^{-N})$. Three classes of contours have been investigated in [34]:

  * •

Parabolic contours

$\varphi(\theta)=N(0.1309-0.1194\theta^{2}+0.2500i\theta)$

  * •

Hyperbolic contours

$\varphi(\theta)=2.246N(1-\sin(1.1721-0.3443i\theta))$

  * •

Cotangent contours

$\varphi(\theta)=N(0.5017\theta\cot(0.6407\theta)-0.6122+0.2645i\theta)$

Note in each case the dependence of the family on $N$.

The procedure for computing the kernel in equation (5.15) is first compute the
function $\Gamma_{d+1,d+1}^{c_{s}(z)\gamma}(t)$ by utilising the explicit
formula (5.9) for piecewise linear paths. By taking

$f(z)=z^{-1}\Gamma_{d+1,d+1}^{c_{s}(z)\gamma}(t)$

we can approximate the contour integral by one of the approaches described
above.

###  5.4. Expected signatures for general kernels

The representation of the previous subsection can be combined with the ideas
of Section 3 to obtain similar representations for
$\left\langle\mathbb{E}\left[S\left(\circ
B\right)\right],S\left(\gamma\right)\right\rangle_{\phi}$ for general $\phi$
satisfying the conditions of Theorem 3.11. The expression is as follows.

######  Theorem 5.5.

Let $\mu$ be a finite signed Borel measure $\mu$ on $\mathbb{R}$. Suppose that
$\phi:\mathbb{N}\cup\\{0\\}\to\mathbb{C}$ is such that

$\phi\left(k\right)=\int_{G}r\left(k,\widetilde{z}\right)\mu\left(d\widetilde{z}\right)\in\mathbb{C}\,\text{,
for all }k\in\mathbb{N}\cup\\{0\\}$

where $r\left(k,\cdot\right)$ is assumed to have the form
$r\left(k,\widetilde{z}\right)=$ $g\left(\widetilde{z}\right)^{\alpha k}$
$\in\mathbb{C}$ for $\alpha\in\mathbb{R}$ and some function
$g:\mathbb{C}\rightarrow\mathbb{C}.$ We assume that $\phi$ satisfies the
conditions in Theorem 3.11, and that $B$ a $d$-dimensional Brownian motion.
Then the expected Stratonovich signature, $\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right]$, belongs to $T_{|\phi|}\left(V\right)$ for any $0\leq
s<\infty$ and

(5.19) $\left|\left|\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right]\right|\right|_{\phi}^{2}=\frac{1}{2\pi
i}\oint_{C}\int_{G}\left[z^{-1}e^{z}\exp\left(\frac{g(\widetilde{z})^{2\alpha}s^{2}d}{4z}\right)\right]\mu(d\widetilde{z})dz$

where $C$ is unit circle in $\mathbb{C}$ traversed anticlockwise. Furthermore
if $\gamma$ is any continuous path of bounded variation it holds that

(5.20)
$K_{\phi}^{\mathcal{W},\gamma}\left(s,t\right):=\left\langle\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right],S\left(\gamma\right)_{0,t}\right\rangle_{\phi}=\frac{1}{2\pi
i}\oint_{C}\int_{G}\left[z^{-1}e^{z}\Gamma_{d+1,d+1}^{c_{g,\alpha,s}(\widetilde{z},z)\gamma}(t)\right]\mu(d\widetilde{z})dz$

where
$c_{g,\alpha,s}(\widetilde{z},z):=g(\widetilde{z})^{\alpha}\sqrt{s/(2z)}\in\mathbb{C}$
and $\Gamma_{d+1,d+1}^{c_{g,\alpha,s}(\widetilde{z},z)\gamma}(t)$ is the
series (5.8), i.e. the last entry of the solution to ODE (5.7).

###### Proof.

The conditions for $\phi$ in Theorem 3.11 and by now standards estimates allow
for the steps of the proof of Theorem 5.4 to be repeated making the obvious
modifications. ∎

As a special case, if $\phi$ is the moments of a random variable $\pi$, i.e.

(5.21) $\phi(k)=\mathbb{E}[\pi^{k}],\ \forall k\geq 0,$

the representations are as follows.

######  Corollary 5.6.

Let the function $\phi:\mathbb{N}\cup\left\\{0\right\\}\rightarrow\mathbb{R}$
as defined in (5.21) and $\psi(k)=\mathbb{E}[|\pi|^{k}]$ such that $\psi$
satisfies Condition 1. Suppose $B$ is a $d$-dimensional Brownian motion, then
the expected Stratonovich signature, $\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right]$, belongs to $T_{|\phi|}\left(V\right)$ for any $0\leq
s<\infty$ and

(5.22) $\left|\left|\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right]\right|\right|_{\phi}^{2}=\frac{1}{2\pi
i}\oint_{C}z^{-1}e^{z}\mathbb{E}_{\pi}\left[e^{(\pi s)^{2}d/(4z)}\right]dz.$

If $\gamma$ is any continuous path of bounded variation it holds that

(5.23)
$K_{\phi}^{\mathcal{W},\gamma}\left(s,t\right)\left(s,t\right):=\left\langle\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right],S\left(\gamma\right)_{0,t}\right\rangle_{\phi}=\frac{1}{2\pi
i}\oint_{C}z^{-1}e^{z}\mathbb{E}_{\pi}\left[\Gamma_{d+1,d+1}^{c_{s}(\pi,z)\gamma}(t)\right]dz$

where $c_{s}(x,z):=x\sqrt{s/(2z)}\in\mathbb{C}$ and
$\Gamma_{d+1,d+1}^{c_{s}(x,z)\gamma}(t)$ is the series (5.8).

As an example, we recall the case
$\phi(k)=\frac{\Gamma(m+1)\Gamma(k+1)}{\Gamma(k+m+1)}$ studied already in
Section 3. Suppose the random variable $\pi\sim\textrm{Beta}(1,m)$ is Beta
distributed, then the moments of $\pi$ are

$\mathbb{E}[\pi^{k}]=\frac{B(k+1,m)}{B(1,m)}=\phi(k).$

We then have the following.

######  Example 5.7.

Let $\phi(k)=\frac{\Gamma(m+1)\Gamma(k+1)}{\Gamma(k+m+1)}$ and $B$ a
$d$-dimensional Brownian motion. Then $\phi$ satisfies Condition 1. The
expected Stratonovich signature, $\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right]$, is well defined and belongs to
$T_{\phi}\left(V\right)$ for any $0\leq s<\infty$, and the squared norm

(5.24) $\left|\left|\mathbb{E}\left[S\left(\circ
B\right)_{0,s}\right]\right|\right|_{\phi}^{2}=\frac{\Gamma(m+1)}{2\pi
i}\oint_{C}z^{-(m+1)}e^{z}\frac{dz}{\sqrt{1-s^{2}d/z^{2}}}$

If $\gamma$ is any continuous path of bounded variation, then

(5.25) $K_{\phi}^{\gamma,W}\left(s,t\right)=\frac{\Gamma(m+1)}{2\pi
i}\oint_{C}z^{-(m+1)}e^{z}\left[\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\Gamma_{d+1,d+1}^{c_{s}(x,z)\gamma}(t)e^{-\frac{x^{2}}{2}}dx\right]dz$

where $c_{s}(x,z)=z^{-1}x\sqrt{s}\in\mathbb{C}$ and
$\Gamma_{d+1,d+1}^{c_{s}(x,z)\gamma}(t)$ is the series (5.8).

The representations above are slightly different from Corollary 5.6 in which
$\pi$ should be a Beta random variable. The expressions above are obtained by
the formulas below:

$\frac{\Gamma(2k+1)}{2^{k}k!}=(2k-1)!!=\mathbb{E}_{X}[X^{2k}]\ \text{and}\
\frac{1}{\Gamma(2k+m+1)}=\oint_{C}z^{-(2k+m+1)}e^{z}dz$

where $X\sim N(0,1)$ is a standard normal random variable. In the point view
of computation, the Gaussian quadrature for approximating the formula (5.25)
is much easier than using the formula (5.23) with $\pi\sim\textrm{Beta}(1,m)$.

######  Remark 5.8.

In terms of the computation procedure, we take the signature kernel in
equation (5.25) as an example. It can be calculated in three successive steps.
First, for fixed $z$, $x$ and $s$, get the exact value of
$\Gamma_{d+1,d+1}^{c_{s}(x,z)\gamma}(t)$ by the explicit solution (5.9) to ODE
(5.7) for piecewise linear path. Second, approximate the expectation

$\mathbb{E}_{X}\left[\Gamma_{d+1,d+1}^{c_{s}(X,z)\gamma}(t)\right]=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\Gamma_{d+1,d+1}^{c_{s}(x,z)\gamma}(t)e^{-\frac{x^{2}}{2}}dx$

by classical Gaussian quadrature on the whole real line. Third, approximate
the contour integral using one of the methods described above. The steps are
summarised schematically as follows:

$K_{\phi}^{\mathcal{W},\gamma}\left(s,t\right)=\frac{\Gamma(m+1)}{2\pi
i}\underbrace{\oint_{C}z^{-(m+1)}e^{z}}_{(3)\ Contour\
approximation}\left[\underbrace{\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\overbrace{\Gamma_{d+1,d+1}^{c_{s}(x,z)\gamma}(t)}^{(1)\
explicit\ solution}e^{-\frac{x^{2}}{2}}dx}_{(2)\ Gaussian\
quadrature}\right]dz.$

The general form (5.20) can also be computed by these three steps successively
but the quadrature formula will generally be more complicated to implement
than the Beta random variable case. See Section 4 for details.

##  6\. Optimal Discrete Measures on Paths

In the previous sections, we have introduced the $\phi$-signature kernels. We
described method for the evaluation of these kernels for a pair of continuous
bounded variation paths, and derived a closed-form expression for the expected
signature against Brownian motion. In particular, given a finite collection of
continuous bounded variation paths
$\\{\gamma_{1},\gamma_{2},\cdots,\gamma_{n}\\}$ on $V$ and a discrete measure
$\mu=\sum_{i=1}^{n}\lambda_{i}\delta_{\gamma_{i}}$supported on this set we can
evaluate

$\left|\left|\mathbb{E}_{X\sim\mu}\left[S\left(X\right)_{0,1}\right]\right|\right|_{\phi}^{2}=\sum_{i,j=1}^{n}\lambda_{i}\lambda_{j}K_{\phi}^{\gamma_{i},\gamma_{j}},$

and also

$\left\langle\mathbb{E}_{X\sim\mathcal{W}}\left[S\left(X\right)_{0,1}\right],\mathbb{E}_{X\sim\mu}\left[S\left(X\right)_{0,1}\right]\right\rangle_{\phi},$

where $\mathcal{W}$ denotes the Wiener measure. This can be used to measure
the similarity of using the maximum mean discrepancy distance associated with
the $\phi$signature kernel:

$d_{\phi}^{2}\left(\mathcal{W},\mu\right)=\left|\left|\mathbb{E}_{X\sim\mathcal{W}}\left[S\left(X\right)_{0,1}\right]-\mathbb{E}_{X\sim\mu}\left[S\left(X\right)_{0,1}\right]\right|\right|_{\phi}^{2},$

which can be used as the basis of goodness-of-fit tests to measure the
similarity of $\mu$ to Wiener measure. We refer to [15] and [8] where kernels
have been proposed as a way to support similar analyses.

Changing our perspective, we can also attempt to find the optimiser over some
subset of measures $C,$.i.e.

(6.1) $\mu^{\ast}=\arg\min_{\mu\in
C}\left|\left|\mathbb{E}_{X\sim\mathcal{W}}\left[S\left(X\right)_{0,1}\right]-\mathbb{E}_{X\sim\mu}\left[S\left(X\right)_{0,1}\right]\right|\right|_{\phi}^{2}$

to give the $d_{\phi}$-best approximation to Wiener measure on $C$. An example
in which this is tractable is when the support of $\mu$ in $C$ is fixed to be
$\\{\gamma_{1},\gamma_{2},\cdots,\gamma_{n}\\}$ and where the set over which
the optimisation is carried our is the set of probability measures with this
support. In other words, $C$ can be identified with the simplex
$C_{n}=\left\\{\lambda:\sum_{i=}^{n}\lambda_{i}=1,\lambda_{i}\geq 0\right\\}$.
By finding this optimum we can then compare the value
$d_{\phi}\left(\mathcal{W},\mu\right)$, for a given measure $\mu,$ to the
optimised value $d_{\phi}\left(\mathcal{W},\mu^{\ast}\right)$ to and use as a
guide to whether $\mu$ is $d_{\phi}$-close to $\mathcal{W}$ when compared to
discrete measures having the same support. A closely related, although more
advanced problem, is the $\phi-$cubature problem of solving

$(\mu^{\ast},\\{\gamma_{i}\\}^{\ast})=\arg\min_{(\mu,\\{\gamma_{i}\\})}\left|\left|\mathbb{E}\left[S\left(\circ
B\right)_{0,1}\right]-\sum_{i=1}^{n}\lambda_{i}S\left(\gamma_{i}\right)_{0,1}\right|\right|_{\phi}^{2},$

which in the case where $\phi\left(n\right)=0$ for $n\geq N$ corresponds to
find a degree$-N$ cubature formula in the sense of [25]. For $N$ large enough
this can be minimised (not necessarily uniquely) to zero and explicit formulas
for $\left(\lambda_{i},\gamma_{i}\right)$ are known in some case; again see
[25] for more details

###  6.1. Existence and Uniqueness of Optimal Discrete Measure

In this subsection, we consider in detail the problem described above. We give
conditions on the collection $\\{\gamma_{1},\gamma_{2},\cdots,\gamma_{n}\\}$
so that

$L\left(\mu\right)=d_{\phi}^{2}\left(\mathcal{W},\mu\right)$

has a unique minimiser on the set

$C_{n}=\left\\{\mu={\textstyle\sum\nolimits_{i=1}^{n}}\lambda_{i}\delta_{\gamma_{i}}:\lambda_{i}\geq
0,\lambda_{1}+...+\lambda_{n}=1\right\\}.$

In order to find the optimal discrete measure on the set of paths
$\\{\gamma_{i}\\}_{i=1}^{n}$, we could solve the problem in equation (6.1)
with constraints $\lambda_{i}\geq 0$ and $\sum_{i=1}^{n}\lambda_{i}=1$. This
is equivalent to solving the quadratic optimisation problem of quadratic
functions with linear equality and inequality constraints given by

(6.2) $\begin{split}&\min_{x\in\mathbb{R}^{n}}\frac{1}{2}x^{T}Kx-h^{T}x\\\
&\text{subject to}\ \mathbf{1}^{T}x=1,\ x\geq 0.\end{split}$

where

$K=\left(K_{\phi}^{\gamma_{i},\gamma_{j}}\right)_{i,j=1,\cdots,n},\text{ and
}h=\left(K_{\phi}^{\gamma_{1},\mathcal{W}},\cdots,K_{\phi}^{\gamma_{n},\mathcal{W}}\right)^{T}.$

Existence and uniqueness of the optimal solution is guaranteed by the positive
definiteness of $K$. Some sufficient conditions for positive definiteness can
be obtained from the following lemma.

######  Lemma 6.1.

The set of all signatures $\mathcal{S}$ of continuous bounded variation paths
is a linearly independent subset of $T\left(\left(V\right)\right).$

###### Proof.

Suppose that $\left\\{h_{1},...,h_{n}\right\\}$ is a subset of $\mathcal{S}$
and suppose that $\sum_{i=1}^{n}\lambda_{i}h_{i}=0$ with not all
$\lambda_{i}=0,$ e.g., suppose that $\lambda_{j}\neq 0.$ The vectors
$h_{1},...h_{n}$ are distinct and so there exist linear functionals $f_{i}$ on
$T\left(\left(V\right)\right)$ for $i\neq j$ with $f_{i}\left(h_{i}\right)=0$
and $f_{j}\left(h_{j}\right)=1.$ Let
$p:T\left(\left(V\right)\right)\rightarrow\mathbb{R}$ be the polynomial
$p\left(x\right)={\textstyle\prod\nolimits_{i\neq j}}f_{i}\left(x\right)$ then
the linear functional $L$ defined by the shuffle product $L=f_{1}\shuffle
f_{2}...\shuffle f_{n}$ agrees with $p$ on $\mathcal{S}$ and hence we arrive
at the contradiction

$\lambda_{j}=\sum_{i=1}^{n}\lambda_{i}L\left(h_{i}\right)=0.$

∎

######  Corollary 6.2.

Let $\left\\{\gamma_{1},...,\gamma_{n}\right\\}$ be a collection of continuous
$V-$valued paths of bounded variation having distinct signatures. If
$\phi:\mathbb{N\cup}\left\\{0\right\\}\rightarrow\left(0,\infty\right)$
satisfies Condition 1 then the matrix $K=(\left\langle
S\left(\gamma_{i}\right),S\left(\gamma_{j}\right)\right\rangle_{\phi})_{i,j=1,...,n}$
is positive definite.

###### Proof.

If $0\neq x\in\mathbb{R}^{n}$ then the previous proposition ensures that
$\sum_{i=1}^{n}x_{i}S\left(\gamma_{i}\right)_{a,b}\neq 0.$ Since
$\left|\left|\cdot\right|\right|_{\phi}$ is a norm we have

$0<\left|\left|\sum_{i=1}^{n}x_{i}S\left(\gamma_{i}\right)_{a,b}\right|\right|_{\phi}^{2}=x^{T}Kx$

as required. ∎

We now prove an existence and uniqueness theorem for the closest discrete
probability measure to Wiener measure which is supported on
$\left\\{\gamma_{1},...,\gamma_{n}\right\\}.$

######  Proposition 6.3.

Let $\left\\{\gamma_{1},...,\gamma_{n}\right\\}$ be a collection of continuous
$V-$valued paths of bounded variation defined over $\left[a,b\right]$ and
having distinct signatures. Assume that
$\phi:\mathbb{N\cup}\left\\{0\right\\}\rightarrow\left(0,\infty\right)$
satisfies Condition 1. Let $C_{n}$ denote the $n-$simplex
$\left\\{\mu=\left(\mu_{1},...,\mu_{n}\right):\sum_{i=1}^{n}\mu_{i}=1,\mu_{i}\geq
0\right\\}$ so that $C_{n}$ is in one-to-one correspondence with the set of
probability measures supported on $\left\\{\gamma_{1},...,\gamma_{n}\right\\}$
by the identification of $\mu$ with
$\sum_{i=1}^{n}\mu_{i}\delta_{\gamma_{i}}.$ Then there exists a unique
$\mu^{\ast}$ which minimises $d_{\phi}\left(\mu,\mathcal{W}\right)$ over $\mu$
in $C_{n},$i.e.

$\mu^{\ast}=\arg\min_{\mu\in C_{n}}\left|\left|\mathbb{E}\left[S\left(\circ
B\right)_{a,b}\right]-\mathbb{E}_{X\sim\mu}\left[S\left(X\right)_{a,b}\right]\right|\right|_{\phi}$

###### Proof.

It is easy to verify that the set $C_{n}$ is a compact and convex set in
$\mathbb{R}^{n}$. Since $f(x)=\frac{1}{2}x^{T}Kx-h^{T}x$ is continuous on the
compact set $C_{n}$, then $f$ is bounded and attains its minimum on some
points in the set $C_{n}$. That means that there exist optimal solutions
$x^{*}\in C_{n}$ such that

$f(x^{*})=\min_{x\in C_{n}}f(x).$

Let $m=\min_{x\in C_{n}}f(x)$ and $x_{1}^{*}$, $x_{2}^{*}\in C_{n}$ be two
optimal solutions. Then, for any $\alpha\in[0,1]$, we have

$\alpha x_{1}^{*}+(1-\alpha)x_{2}^{*}\in C_{n}$

and

$m\leq f(\alpha x_{1}^{*}+(1-\alpha)x_{2}^{*})\leq\alpha
f(x_{1}^{*})+(1-\alpha)f(x_{2}^{*})=m.$

Thus,

$\frac{1}{2}(x_{1}^{*})^{T}Kx_{2}^{*}-\frac{1}{2}h^{T}(x_{1}^{*}+x_{2}^{*})=m.$

Since

$\displaystyle
f(x_{1}^{*})=\frac{1}{2}(x_{1}^{*})^{T}Kx_{1}^{*}-h^{T}x_{1}^{*}=m\
\text{and}\
f(x_{2}^{*})=\frac{1}{2}(x_{2}^{*})^{T}Kx_{2}^{*}-h^{T}x_{2}^{*}=m,$

combining above three equations together, we have

$(x_{2}^{*}-x_{1}^{*})^{T}K(x_{2}^{*}-x_{1}^{*})=0.$

Since the matrix $K$ is positive definite on $\mathbb{R}^{n}$, we must have
that $x_{1}^{*}=x_{2}^{*}$. So we have concluded our proof. ∎

######  Remark 6.4.

The next aim is to find the optimal measure in Theorem 6.3 and the minised
value of the objective. In some cases this can be done explicitly. Letting $f$
be the function in the proof, we have the following cases:

Case 1:

There exists $x^{\ast}\in C_{n}$ such that $\nabla f(x^{\ast})=0$. Then the
optimal solution and the value are

$x^{\ast}=K^{-1}h\in C_{n},\ f(x^{\ast})=-\frac{1}{2}h^{T}K^{-1}h.$

Case 2:

Assume that $\nabla f$ is non-vanishing on $C_{n}$. If there exists a vertex
$e_{m}$ of $C_{n}$ such that $f\left(e_{m}\right)<f\left(e_{j}\right)$ for all
$j\neq m$ and if it satisfies that

(6.3) $(Ke_{m}-h)^{T}(e_{i}-e_{m})\geq 0,\ \forall
i\in[n]:=\\{1,2,\cdots,n\\},$

then the optimal solution is $e_{m}$ and
$f(e_{m})=\frac{1}{2}e_{m}^{T}Ke_{m}-h^{T}e_{m}$. Actually, we have

$\displaystyle f(x)-f(e_{m})$ $\displaystyle=\nabla
f(e_{m})^{T}(x-e_{m})+\frac{1}{2}(x-e_{m})^{T}\nabla^{2}f(e_{m})(x-e_{m})$
$\displaystyle=(Ke_{m}-h)^{T}(x-e_{m})+\frac{1}{2}(x-e_{m})^{T}K(x-e_{m})$
$\displaystyle=(Ke_{m}-h)^{T}\left(\text{$\sum$}_{i=1}^{n}\alpha_{i}e_{i}-e_{m}\right)+\frac{1}{2}(x-e_{m})^{T}K(x-e_{m})$
$\displaystyle=\text{$\sum$}_{i=1}^{n}\alpha_{i}(Ke_{m}-h)^{T}(e_{i}-e_{m})+\frac{1}{2}(x-e_{m})^{T}K(x-e_{m})$
$\displaystyle\geq 0,$

where $x=\text{$\sum$}_{i=1}^{n}\alpha_{i}e_{i}$ is a convex combination of
vertexes of $C_{n}$. The condition (6.3) means that

$\displaystyle\tilde{f}(t)$ $\displaystyle=f((1-t)e_{m}+te_{i})$
$\displaystyle=\frac{1}{2}(e_{i}-e_{m})^{T}K(e_{i}-e_{m})t^{2}+(Ke_{m}-h)^{T}(e_{i}-e_{m})t+f(e_{m})$

is increasing on the interval $[0,1]$.

If $\nabla f$ does not vanish in $C_{n}$ and the conditions in case 2 of the
above do not hold, then there is no explicit expression for the optimal
solution and alternative numerical methods are needed to determine the
minimiser. Common tools are active-set methods and interior point methods; see
[37, 38] and the references therein).

##  7\. Examples and Numerical Results

In this section, we give some numerical results to illustrate the usefulness
of general signature kernels in measuring the similarity/alignment between a
given discrete measures on paths and Wiener measure. We illustrate the use of
these measures in a number of examples. As in the previous section let
$\mu=\sum_{i=1}^{n}\lambda_{i}\delta_{\gamma_{i}}$ be a discrete probability
measure supported on a finite collection of continuous bounded variation paths
$\gamma:[0,1]\rightarrow V$ and denote the Wiener measure on $\mathcal{W}$. A
plausible measure of the alignment between these two expected signatures is

(7.1)
$\cos\angle_{\phi}(\mu,\mathcal{W}):=\frac{\left\langle\mathbb{E}_{X\sim\mu}\left[S\left(\circ
X\right)_{0,1}\right],\mathbb{E}_{X\sim\mathcal{W}}\left[S\left(X\right)_{0,1}\right]\right\rangle_{\phi}}{\left|\left|\mathbb{E}_{X\sim\mu}\left[S\left(\circ
X\right)_{0,1}\right]\right|\right|_{\phi}\left|\left|\mathbb{E}_{X\sim\mathcal{W}}\left[S\left(X\right)_{0,1}\right]\right|\right|_{\phi}}.$

It follows from our earlier discussion that
$\cos\angle_{\phi}(\mu,\mathcal{W})\in\left[0,1\right].$ A justification for
this quantity measuring the alignment of the measures $\mu$ and $\mathcal{W}$,
rather than just their expected signatures, is that for any given pair of
measures $\nu_{1}$ and $\nu_{2}$ on a space of (rough) paths it holds that
$\cos\angle_{\phi}(\nu_{1},\nu_{2})=1$ if and only if there exists
$\lambda\in\mathbb{R}$ with

$\mathbb{E}_{X\sim\nu_{1}}\left[S\left(\circ
X\right)_{0,1}\right]=\lambda\mathbb{E}_{X\sim\nu_{2}}\left[S\left(\circ
X\right)_{0,1}\right].$

The fact that $\lambda=1,$ and hence that the expected signatures coincide,
follows by interpreting this equality under the projection
$\pi_{0}:T\left(\left(V\right)\right)\rightarrow\mathbb{R}$. Another quantity
we use is the MMD distance

(7.2)
$d_{\phi}(\mu,\mathcal{W})=\left|\left|\mathbb{E}_{X\sim\mu}\left[S\left(\circ
X\right)_{0,1}\right]-\mathbb{E}_{X\sim\mathcal{W}}\left[S\left(X\right)_{0,1}\right]\right|\right|_{\phi},$

which we have already discussed extensively.

###  7.1. Discrete Measures on Brownian Paths

In Section 6, we proved the existence of a unique optimal probability measure
$\mu^{\ast}$ supported on $\\{\gamma_{1},\cdots,\gamma_{n}\\}$ such that

$\mu^{\ast}=\arg\min_{\mu\in C_{n}}\left|\left|\mathbb{E}\left[S\left(\circ
B\right)_{0,1}\right]-\mathbb{E}^{\mu}\left[S\left(\gamma\right)_{0,1}\right]\right|\right|_{\phi}^{2}.$

We now present an example in which $\\{\gamma_{1},\cdots,\gamma_{n}\\}$ is
obtained as the piecewise linear interpolation of $n$ i.i.d discretely-sampled
Brownian paths. We consider two cases for $\phi:$

  1. (1)

$\phi(k)=\left(\frac{k}{2}\right)!$ for
$n\in\mathbb{N}\cup\left\\{0\right\\}.$We refer to the resulting
$\phi$-signature kernel, somewhat inexactly, as the the factorially-weighted
signature kernel.

  2. (2)

The original signature kernel $\phi(k)\equiv 1$.

######  Example 7.1.

We randomly sample $n$ i.i.d. Brownian motion paths in $\mathbb{R}^{d}$. Each
path sampled over the time interval $[0,1]$, on an equally-spaced partition
$0=t_{1}<t_{2}<\cdots<t_{m}=1$ with $t_{j+1}-t_{j}=\frac{1}{m-1}$. We denote
the resulting finite set piecewise linearly interpolated Brownian sample paths
as

$\mathcal{S}(n,m,d)=\\{B_{i}\\}_{i=1}^{n}\ \text{with}\
B_{i}=\\{B_{i}(t_{j})\in\mathbb{R}^{d}\\}_{j=1}^{m}.$

Figure 7.1 and 7.2, displays the alignment
$\cos\angle_{\phi}(\mu^{*},\mathcal{W})$ and the similarity
$d_{\phi}(\mu^{*},\mathcal{W})$ for the optimal discrete probability measure
supported on $\mathcal{S}(n,m,d)$, in which the number of sample paths $n=10$
and the observation points $m=10$ are fixed and the dimension $d$ is varied
over the range 2 to 6. We run 400 independent experiments for each $d$, that
is, we generate 400 independent samples of the sets $\mathcal{S}(n,m,d)$ for
each dimension $d$. Each set $\mathcal{S}(n,m,d)$ has an optimal measure
associated with it, which we compute. The boxplots in Figure 7.1 and 7.2 show
the median, range and interquartile range of the values of the alignment and
the similarity of the optimal discrete measures over these 400 samples.
Qualitatively we can see from both quantities show dependence on the dimension
of the state space, with the alignment decreasing and the dis-similarity
increasing w.r.t. the dimension. We can also compare the results using the two
different $\phi$-signature kernels with the original signature kernel showing
the same behaviour w.r.t. the dimension having a persistently higher level of
alignment than under the factorially-weighted signature kernel across all of
the dimensions considered.

Figure 7.1. Boxplots of the factorially-weighted signature kernel. (a) The
left panel shows the distribution of the values of the alignment
$\cos\angle_{\phi}(\mu^{*},\mathcal{W})$ of the optimal measure and the Wiener
measure across 400 samples. The x-axis is the dimension of the Brownian
motion, and the y-axis the value of the alignment. (b) The right panel shows
the same for the MMD distance $d_{\phi}(\mu^{*},\mathcal{W}).$ Figure 7.2. The
optimal measure under the original signature kernel

###  7.2. Examples using cubature formulae

In the paper [25], Lyons and Victoir studied cubature on Wiener space. Let
$C_{bv}([0,T],V)$ be a subset of Wiener space made of bounded variation paths.
We say that the paths $\gamma_{1},\cdots,\gamma_{n}\in C_{bv}([0,T],V)$ and
the positive weights $\lambda_{1},\cdots,\lambda_{n}$ define a cubature
formula on Wiener space of degree $m$ at time $T$ if

$\mathbb{E}\left[S\left(\circ
B\right)_{0,T}(e_{I}^{*})\right]=\sum_{j=1}^{n}\lambda_{j}S\left(\gamma_{j}\right)_{0,T}(e_{I}^{*})$

for all $I\in\mathcal{A}_{m}:=\\{I=(i_{1},\cdots,i_{k}):k\leq m\\}$ with
$m\in\mathbb{N}$.

Cubature on Wiener space can be an effective way to develop high-order
numerical schemes for high-dimensional stochastic differential equations and
parabolic partial differential equations, see [25]. In Section 5 of [25], the
authors also construct an explicit cubature formula of degree 5 for
2-dimensional Brownian motion. The reader can find formulas of these cubature
paths and measure in tables 2 and 3 in the same reference.

In this subsection, we analyse the results for a family of $\phi$-signature
kernels on three discrete probability measures supported on this collection of
cubature paths. We consider the cubature weights themselves, the empirical
measure of the sample (i.e. where they are equally weighted) and the optimal
measure obtained from Section 6. In Figure 7.3, we show the similarity of
these discrete measures and the Wiener measure under the family of Beta-
weighted signature kernels given by

(7.3) $\phi(k)=\frac{\Gamma(m+1)\Gamma(k+1)}{\Gamma(k+m+1)}$

for various values of $m$ in the weight $\phi$ (shown along the horizontal
axis).

The plot on the left panel of 7.3 shows that as the parameter $m$ increases
these three distances first increase fast and then gradually go down. We see
that the distance of the optimal measure and the Wiener measure is smallest
and the distance of the empirical measure is much larger than the distance of
cubature measure. The right pane shows the ratio of the distance of optimal
measure and the distance of cubature measure for different choices of $m$.

Figure 7.3. The similarity under a family of Beta-weighted signature kernels.
The left panel is the plot of the distance of these discrete measures and the
Wiener measure plotted against different values of $m$ on the horizontal axis.
The right panel plots the ratio of the optimal distance and the cubature
distance.

###  7.3. Applications in Signal Processing

The alignment in equation (7.1) and the similarity in equation (7.2) defined
by the $\phi$-signature kernel give us a way of determining how large a given
discrete measure is different to the Wiener measure. We can use these
quantities to measure deviation of a discrete measure from a reference measure
(i.e. the Wiener measure here). A natural application of these methods in
signal processing is to mitigate/detect the (additive) contamination of white
noise under different types of perturbation.

The examples studied here are motivated by an attempt to study radio frequency
interference (RFI) in the radio astronomy. In this setting astronomers would
like to obtain high-resolution sky images of an interested astrophysical
object using measurements from an array of antennas (e.g. the Karl G. Jansky
Very Large Array (VLA) etc.). To observe the sky and then synthesis the sky
image interested. The observation is called visibility $V_{ij}(t,v,p)$, where
$ij$ is an antenna pair, $t$ is the time integration, $v$ is the frequency and
$p$ is the polarization. Usually the visibility would be contaminated by
thermal noise and radio frequency interference (RFI). So the observation data
from an interferometer can be broken down into three components: the
astrophysical sky signals, thermal noise and RFI. The first component is
slowly varying which can be removed in the observation data by sky-subtraction
method (see e.g. [36]). The RFI signal is usually much stronger than thermal
noise but is also sometimes ultra-faint. For different antennas, the RFI
contamination is systematic and thermal noise can be assumed to be
independent. In order to obtain a high-resolution image, the first step is to
design some methods to identify and then, if possible, to remove the RFI
component of the observation.

We consider two idealised types of RFI contamination. The first is by simple
superposition with a sine wave of a fixed single frequency and a given
amplitude and phase, so that the interference is narrow-band but persistent
over time. The second will be to consider a short duration spike, as modelled
in the paper by Davis and Monroe [10] in the univariate setting, in which the
Brownian signal undergoes a perturbation at a uniformly distributed random
time to give

(7.4) $B(t)+\epsilon\sqrt{(t-U)^{+}}.$

We again compare the use of two $\phi$-signature kernels. The factorially-
weighted signature kernel and the original signature kernel.

######  Example 7.2.

Working in $d$-dimensions we take a path of the form

$X_{i}^{(j)}(t)=B_{i}^{(j)}(t)+\epsilon\sin(2\pi\nu t-\phi_{i}^{(j)}),\
j=1,2,\cdots,d$

where the frequency $\nu$ is fixed, the phase shifts are $\phi_{i}^{(j)}$ and
$\epsilon$ denotes a (small) fixed amplitude. Let a finite collection of
sample paths on time interval $[0,1]$ as

$\mathcal{S}(n,m,d)=\\{X_{i}\\}_{i=1}^{n},\ \text{where}\
X_{i}=\\{(X_{i}^{(1)}(t_{j}),\cdots,X_{i}^{(d)}(t_{j}))\in\mathbb{R}^{d}\\}_{j=1}^{m}.$

In Figure 7.4 and 7.5, we fixed $(n,m,d)=(10,10,2)$, $\epsilon$ from
$[0,1]$and the frequency $\nu\in\\{2,3\\}$. We run 100 collections of paths
$\mathcal{S}(n,m,d)$ for each $\epsilon$ and frequency $\nu$. The figures show
the deviation of the alignment and the similarity of the optimal measure (the
empirical measure, resp.) and the Wiener measure, in which the middle line is
the median of the alignment or the similarity resp., and the shadow represents
the range from the lower quartile to the upper quartile. We generate 100
experiments for each $\epsilon$. The figures show that the alignment decreases
very fast to a low level and the dis-similarity increases very quickly as
$\epsilon$ becomes large for both the optimal measure and the empirical
measure. At larger frequencies $\nu$, the alignment (dis-similarity) decays
(grows) more rapidly.

Finally we present an example based on the construction in the paper of Davis
and Monroe [10] mentioned earlier. Here the interference is characterised by a
sudden high energy spike at a uniform random time.

Figure 7.4. The case for the factorially-weighted signature kernel. (a) and
(c) show similarities of discrete measures and the Wiener measure where the
horizontal is the value of $\epsilon$ and vertical axis is the value of
alignment. (b) and (d) show similarities of discrete measures and the Wiener
measure. The solid line is for the optimal measure while the dashed line is
for the empirical measure. The upper panel is for the frequency $\nu=2$ and
the lower is for $\nu=3$. Figure 7.5. The same example under the original
signature kernel

######  Example 7.3.

We define

$X_{i}^{(j)}(t)=B_{i}^{(j)}(t)+\epsilon\sqrt{(t-U_{i})^{+}},\ j=1,2,\cdots,d$

where $U_{i}$ is uniformly distributed in $[0,1]$, the time interval
$t\in[0,1]$ and $x^{+}=\max\\{0,x\\}$. We denote a finite collection of these
paths as

$\mathcal{S}(n,m,d)=\\{X_{i}\\}_{i=1}^{n},\ \text{where}\
X_{i}=\\{(X_{i}^{(1)}(t_{j}),\cdots,X_{i}^{(d)}(t_{j}))\in\mathbb{R}^{d}\\}_{j=1}^{m}.$

In Figure 7.6 and 7.7, the parameters $(n,m,d)=(10,10,2)$ are fixed and
$\epsilon$ is taken from $[0,5]$. We run 100 independent experiments for each
$\epsilon$. The plots are like ones in the above example. The middle line is
the median of the alignment (the similarity, resp.) and the shadow is the
range from the lower quartile to the upper quartile of the alignment (the
similarity, resp.) for the 100 collections of sample paths. We can see from
these figures that the alignment (the dis-similarity, resp.) is decreasing
(increasing, resp.) as $\epsilon$ increases, as one would expect. From the
point view of RFI mitigation, the alignment of the empirical measure is more
relevant than that of the optimal measure. It is reasonable that as the
strength $\epsilon$ is large the empirical measure is less similar w.r.t. the
Wiener measure than the optimal measure. The alignment of the empirical
measure decays faster than that of optimal measure in our experiments. This
suggests potential uses for building method for the identification of RFI
based on a threshold for the alignment of the empirical measure. The
preliminary results here for instance suggest that a threshold of alignment of
0.2 under the factorially-weighted signature kernel could be used in this
example.

Figure 7.6. The case for the factorially-weighted signature kernel. (a) The
left panel shows the alignment of discrete measures and the Wiener measure for
each $\epsilon$ taken from $[0,1]$ where x-axis is the value of $\epsilon$ and
y-axis is the value of alignment. (b) The right panel shows the similarity of
discrete measures and the Wiener measure as in (a). The solid line is for the
optimal measure while the dash line is for the empirical measure. Figure 7.7.
The same example under the original signature kernel

## Acknowledgements

The authors thank Cris Salvi and Bojan Nikolic respectively for discussions
related to the Signature Kernel and the problem of RFI mitigation in
radioastronomy.

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