Expected Signature Kernel

• Expected Signature Kernel is a kernel on path space that uses iterated-integral signatures to represent and compare probability measures over paths.
• It bridges deterministic and stochastic models by averaging path signatures, ensuring positive-definiteness and invariance under reparameterization.
• The framework supports scalable numerical approximations and PDE formulations, enabling efficient computation for high-dimensional and discrete datasets.

The expected signature kernel is a kernel on path space or on probability measures over path space built from signatures, the iterated-integral coordinates of paths in the tensor algebra. In its law-space form, it is the inner product of mean signatures, $$K(P,Q)=\langle \mathbb E_{X\sim P}[S(X)],\mathbb E_{Y\sim Q}[S(Y)]\rangle$$, and therefore inherits positive-definiteness from the Hilbert structure of the tensor algebra (Tóth, 21 Jun 2025). Closely related work studies kernels obtained by averaging randomised developments of deterministic paths into matrix Lie groups; in the large-$N$ limit, these constructions recover the ordinary signature kernel for $GL(N)$ and yield a unitary analogue described by free probability and a quadratic functional equation (Cass et al., 2024).

1. Signature-theoretic foundation

Let $V\simeq \mathbb R^d$ and let $\gamma:[a,b]\to V$ be a continuous path of finite $1$-variation. Its signature $S(\gamma)\in T((V))$ is defined by the controlled differential equation

$$d z_t = z_t \otimes d\gamma_t,\qquad z_a = 1\in T((V)),$$

with $S(\gamma)=z_b$. Equivalently, the degree-$m$ coordinate is the iterated integral

$N$0

(Cass et al., 2024).

The classical signature kernel on path space is the Hilbert–Schmidt inner product

$N$1

In the Hilbert-space formulation, if $N$2 is a real Hilbert space and $N$3 is of bounded variation, then each $N$4 is an “ordered” analogue of the $N$5-th sample moment of the infinitesimal increments, and

$N$6

(Király et al., 2016).

This kernel is manifestly positive-definite because it is an inner product in feature space. For bounded-variation paths, the associated RKHS is universal on compact sets in the sense described via Stone–Weierstrass, and the signature is invariant to reparameterization up to tree-like equivalence; augmenting a path by time, $N$7, restores sensitivity to explicit timing when required (Király et al., 2016, Tóth, 21 Jun 2025).

A weighted generalisation replaces the unweighted tensor inner product by

$N$8

for nonnegative weights $N$9 satisfying the factorial-type summability condition

$GL(N)$0

The two-parameter version $GL(N)$1 is central in PDE-based formulations (Cass et al., 2021).

2. From deterministic paths to probability measures

For probability distributions $GL(N)$2 on $GL(N)$3, the Expected Signature Kernel is defined by

$GL(N)$4

Writing $GL(N)$5 and $GL(N)$6, this becomes

$GL(N)$7

In discrete time, with a static kernel lift $GL(N)$8 and truncation $GL(N)$9, one obtains the finite-level approximation

$V\simeq \mathbb R^d$0

(Tóth, 21 Jun 2025).

This law-space kernel is closely tied to kernel-MMD constructions. For two measures $V\simeq \mathbb R^d$1, the signature-MMD satisfies

$V\simeq \mathbb R^d$2

with $V\simeq \mathbb R^d$3, and at level $V\simeq \mathbb R^d$4,

$V\simeq \mathbb R^d$5

(Friz et al., 9 Sep 2025, Alonso, 27 Jun 2026).

A terminological point is important. Some treatments distinguish the expectation of the pathwise signature kernel,

$V\simeq \mathbb R^d$6

from the inner product of expected signatures,

$V\simeq \mathbb R^d$7

The latter is the standard law-space expected signature kernel in the definitions above (Alonso, 27 Jun 2026).

The expected signature itself is not generally group-like. If $V\simeq \mathbb R^d$8 is a random weakly-geometric path, then

$V\simeq \mathbb R^d$9

Accordingly, averaging introduces a covariance defect in the shuffle identity, and $\gamma:[a,b]\to V$0 is group-like if and only if $\gamma:[a,b]\to V$1 is almost surely deterministic, described there as a single tree-like path (Alonso, 27 Jun 2026). A common misconception is therefore to treat the mean signature as though it retained the exact multiplicativity of each individual realisation; the covariance identity shows that this fails except in the degenerate case.

3. Randomised developments and universal large-$\gamma:[a,b]\to V$2 limits

A second major construction starts from random developments of a deterministic path into a matrix Lie group. One chooses a random linear map $\gamma:[a,b]\to V$3, writes $\gamma:[a,b]\to V$4, and solves the matrix-valued CDE

$\gamma:[a,b]\to V$5

with $\gamma:[a,b]\to V$6 or $\gamma:[a,b]\to V$7. This yields feature maps

$\gamma:[a,b]\to V$8

and after averaging over a law $\gamma:[a,b]\to V$9 on $1$0, kernels

$1$1

(Cass et al., 2024).

For $1$2, under very mild moment and weak-dependence assumptions on the entries of $1$3,

$1$4

independently of the precise distribution $1$5. In this sense, random $1$6-based kernels converge to the ordinary signature kernel in the $1$7 limit (Cass et al., 2024).

In the unitary case, the large-$1$8 limit

$1$9

exists. If $S(\gamma)\in T((V))$0, then

$S(\gamma)\in T((V))$1

where $S(\gamma)\in T((V))$2 are freely-independent semicircular operators in a non-commutative probability space $S(\gamma)\in T((V))$3 (Cass et al., 2024).

The key structural conclusion is that the limiting kernels depend on the choice of Lie group but are universal limits with respect to how the development map is randomised. This places the ordinary signature kernel and the unitary free-probability kernel in a common large-$S(\gamma)\in T((V))$4 framework (Cass et al., 2024).

4. Free probability, integral equations, and PDE formulations

For unitary developments, the free-semicircle moment functional $S(\gamma)\in T((V))$5 satisfies Schwinger–Dyson relations. These translate into a quadratic functional equation for

$S(\gamma)\in T((V))$6

namely

$S(\gamma)\in T((V))$7

with boundary conditions $S(\gamma)\in T((V))$8. Setting $S(\gamma)\in T((V))$9, $$d z_t = z_t \otimes d\gamma_t,\qquad z_a = 1\in T((V)),$$0 recovers the full kernel $$d z_t = z_t \otimes d\gamma_t,\qquad z_a = 1\in T((V)),$$1 (Cass et al., 2024).

This integral-equation viewpoint parallels earlier PDE formulations for signature kernels. In the unweighted case $$d z_t = z_t \otimes d\gamma_t,\qquad z_a = 1\in T((V)),$$2, the two-parameter kernel

$$d z_t = z_t \otimes d\gamma_t,\qquad z_a = 1\in T((V)),$$3

is the unique smooth solution of the Goursat PDE

$$d z_t = z_t \otimes d\gamma_t,\qquad z_a = 1\in T((V)),$$4

The weighted theory also admits integral-transform and randomisation representations: if $$d z_t = z_t \otimes d\gamma_t,\qquad z_a = 1\in T((V)),$$5, then

$$d z_t = z_t \otimes d\gamma_t,\qquad z_a = 1\in T((V)),$$6

so computing $$d z_t = z_t \otimes d\gamma_t,\qquad z_a = 1\in T((V)),$$7 reduces to averaging PDE solutions for rescaled paths (Cass et al., 2021).

For stochastic models, analogous PDEs appear at the level of expected signatures. For inhomogeneous Lévy processes with absolutely continuous characteristics, the expected signature is the free development of the characteristic velocity

$$d z_t = z_t \otimes d\gamma_t,\qquad z_a = 1\in T((V)),$$8

and the kernel $$d z_t = z_t \otimes d\gamma_t,\qquad z_a = 1\in T((V)),$$9 satisfies a hyperbolic Goursat-type PDE coupled to two linear ODEs. In the special case of zero-drift continuous Gaussian martingales with deterministic covariation rates $S(\gamma)=z_b$0 and $S(\gamma)=z_b$1, this closes to

$S(\gamma)=z_b$2

(Friz et al., 9 Sep 2025).

The Brownian case admits further closed-form structure. The expected Stratonovich signature of a $S(\gamma)=z_b$3-dimensional Brownian motion over $S(\gamma)=z_b$4 is

$S(\gamma)=z_b$5

and for the factorial weighting $S(\gamma)=z_b$6,

$S(\gamma)=z_b$7

where $S(\gamma)=z_b$8 is the hyperbolic distance of the Cartan development of the rescaled path in the hyperboloid model (Cass et al., 2021).

5. Discretisation, dynamic programming, and scalable approximation

For discrete sequences, the signature kernel admits efficient dynamic programming. If $S(\gamma)=z_b$9 and $m$0 are observed sequences with increments $m$1 and $m$2, then the discrete signature is

$m$3

and the truncated kernel can be computed by a Horner-type recursion in $m$4 time, instead of the direct $m$5 cost of the double sum over subsequences (Király et al., 2016).

The unitary large-$m$6 kernel from the Schwinger–Dyson equation also admits a direct discretisation that avoids computing full signatures. If $m$7 is a partition and $m$8 is a piecewise-constant or piecewise-linear approximation, then on the grid $m$9 the left-point Riemann–Stieltjes recursion is

$N$00

Because each anti-diagonal depends only on earlier anti-diagonals, the scheme can be filled in $N$01 time and parallelised on GPU. Its mesh-size error satisfies

$N$02

so the discrete solution converges uniformly to the true kernel as the mesh tends to zero (Cass et al., 2024).

For large datasets, Random Fourier Signature Features provide an unbiased approximation to truncated signature kernels. With a translation-invariant static kernel $N$03 and independent Random Fourier maps $N$04, the random kernel $N$05 satisfies

$N$06

and under Bernstein or sub-Gaussian assumptions one obtains uniform approximation guarantees over bounded-variation sequences (Toth et al., 2023). In complexity terms, exact dynamic-programming kernel evaluation requires $N$07 time for a Gram matrix of $N$08 sequences of maximum length $N$09, whereas the feature-based variants reduce the $N$10-dependence to $N$11 and replace the $N$12 Gram-matrix cost by an $N$13 primal cost (Toth et al., 2023). For the expected signature kernel on distributions, feature maps $N$14 can be built by averaging pathwise random signature features, yielding $N$15 with uniform high-probability bounds that decay exponentially in the sketch dimension $N$16 (Tóth, 21 Jun 2025).

6. Stochastic models, examples, and uses

For Gaussian processes with strictly regular kernels, the expected signature admits an explicit pairing formula. If

$N$17

has covariance

$N$18

then for any even multi-index $N$19,

$N$20

This induces a positive-definite kernel

$N$21

between Gaussian processes (Boedihardjo et al., 2013).

Fractional Brownian motion with Hurst index $N$22 is a concrete example. With Volterra kernel $N$23, the corresponding density is

$N$24

and the expected signature coordinates are given by the same pairing formula with this choice of $N$25 (Boedihardjo et al., 2013).

The expected signature kernel is also used as a covariance on the space of probability measures. Given training distributions $N$26 and a test distribution $N$27, one may place a Gaussian process prior

$N$28

with posterior mean and covariance determined by the Gram matrix $N$29. Replacing the true expected signatures by empirical averages $N$30 yields a GP on finite samples (Tóth, 21 Jun 2025).

Beyond Gaussian settings, the law-space kernel extends to inhomogeneous Lévy processes via the characteristic-velocity PDE system, offering an analytic alternative to Monte Carlo for signature-MMD computation (Friz et al., 9 Sep 2025). For affine and exponential Hawkes processes, truncated expected signatures admit finite-dimensional linear closures after state-weight augmentation, so the corresponding truncated expected signature kernels can be assembled from matrix exponentials (Alonso, 27 Jun 2026).

Taken together, these formulations show that the expected signature kernel occupies two linked roles. It is, first, a positive-definite kernel on laws of paths built from mean signatures; second, it is connected to pathwise signature kernels through randomisation, free probability, PDEs, and numerical schemes that avoid explicit high-order tensor computation (Tóth, 21 Jun 2025, Cass et al., 2024).