Signature Kernel: Theory, Computation & Applications

• Signature kernel is a universal, positive-definite tool that encodes all iterated-integral features of continuous path data while being invariant to time reparametrization.
• It computes the inner product of tensor signatures via efficient solutions to two-dimensional hyperbolic PDEs, supporting both classical and deep learning methods.
• Its universality and characteristic properties underpin robust MMD tests, proper scoring rules for generative modeling, and scalable approaches for stochastic process analysis.

The signature kernel is a universal, positive-definite kernel function for path-valued data arising from rough path theory, defined as the inner product of the tensor algebraic signatures of two continuous paths. It encodes all iterated-integral (pathwise) features of sequential data, is invariant to time reparametrisation (unless appropriately augmented), and admits efficient computation via solutions to two-dimensional hyperbolic (Goursat) PDEs. Leveraging its strict universality and characteristicness, the signature kernel underpins robust MMD-type metrics, proper scoring rules for generative modeling, and kernel embeddings for stochastic process laws, with scalable implementations supporting both classical and deep learning paradigms.

1. Mathematical Definition and Fundamental Properties

Given a continuous path $X\colon[a, b]\rightarrow\mathbb{R}^d$, its signature $S(X)$ is the sequence of all its iterated integrals: $$S(X)_{a, b} = \left( 1,\, S(X)^i,\, S(X)^{i,j},\,\ldots \right)$$ where, for a multi-index $(i_1,\ldots,i_k)$,

$$S(X)_{a, b}^{i_1 \ldots i_k} = \int_{a < t_k < \cdots < t_1 < b} dX^{i_1}_{t_1} \cdots dX^{i_k}_{t_k}$$

and $S(X)^\emptyset = 1$. The signature encodes all pathwise, temporal, and cross-channel dependencies via its algebraic (shuffle product) and analytic structure.

The signature kernel between two paths $X, Y\colon [a, b] \rightarrow \mathbb{R}^d$ is the Hilbert-space inner product of their (optionally augmented) signatures: $$k(X, Y) = \langle S(\phi(X)), S(\phi(Y)) \rangle_{\mathcal{H}}$$ where $\phi$ denotes an injective augmentation (typically basepoint and time). For practical computation, a static kernel $\kappa$ (e.g. RBF) may be used to lift $\mathbb{R}^d$-valued points into an RKHS prior to signature computation.

Key properties:

• Universality/Characteristicness: The signature feature map (with suitable augmentation and lift) is universal and characteristic to path law, making the kernel uniquely identify probability measures on path space (Chevyrev et al., 2018, Lee et al., 2023).
• Positive-Definiteness: $k(\cdot, \cdot)$ arises as an inner product in Hilbert space, ensuring PD.
• Factorial Decay: Iterated integrals decay factorially in level, making series absolutely convergent for finite 1-variation paths (Lee et al., 2023).
• Algebraic Structure: Signature enjoys the shuffle product, encoding all orderings and interactions (Dodson et al., 21 Oct 2025).
• Robustness: Proper normalization (e.g. tensor scaling, basepoint/time augmentation) yields robustness to tree-like equivalence, reparametrization, and path drift (Chevyrev et al., 2018).

2. Kernel Computation: Goursat PDEs and Fast Algorithms

The computation of the signature kernel is founded on its PDE characterization: for continuously differentiable (or piecewise linear) paths $x, y\colon [0, 1]\to\mathbb{R}^d$, the function

$$k(s, t) := \langle S(x)_{[0, s]}, S(y)_{[0, t]} \rangle$$

solves the Goursat PDE

$$\frac{\partial^2 k}{\partial s\,\partial t}(s, t) = \langle \dot{x}_s, \dot{y}_t \rangle\, k(s, t), \quad k(0, t) = k(s, 0) = 1$$

This PDE admits efficient explicit finite-difference schemes and is well suited for GPU parallelisation by exploiting antidiagonal independence (Salvi et al., 2020, Shmelev et al., 12 Sep 2025).

Extensions include:

• Geometric Rough Paths: The kernel generalizes to the rough signature kernel, defined for geometric $p$-rough paths via the corresponding rough integral Goursat-type equations (Salvi et al., 2020, Lemercier et al., 1 Apr 2024).
• Polynomial and Tilewise Neumann Expansions: For high-roughness inputs, adaptive local expansions achieve high-order accuracy with greatly reduced memory (Tamayo-Rios et al., 27 Feb 2025, Cass et al., 12 Feb 2025).
• Random Feature Approximations: Random Fourier Signature Features and their tensor-projection variants yield unbiased, scalable primal kernel approximators, linear in sequence length and enabling application to extremely large databases (Toth et al., 2023, Tóth et al., 13 Jan 2025).
• Open-source Implementations: State-of-the-art packages (sigkernel, pySigLib, KSig) implement these solvers and feature approaches on modern hardware (Shmelev et al., 12 Sep 2025, Tóth et al., 13 Jan 2025).

3. Theoretical Guarantees: Universality, Characteristicness, Scoring Rules

Given injective augmentation, the signature kernel is universal (dense in $C(K)$ for suitable compact sets $K$) and characteristic (injective when embedding probability measures into the corresponding RKHS) (Chevyrev et al., 2018, Lemercier et al., 1 Apr 2024, Lee et al., 2023, Dodson et al., 21 Oct 2025). Consequently:

• For two measures $\mu, \nu$ on path space,

$$d_k(\mu, \nu) = \| \mathbb{E}_\mu[S(X)] - \mathbb{E}_\nu[S(Y)] \|_{\mathcal{H}}$$

is a metric (Maximum Mean Discrepancy) uniquely identifying laws up to tree-like equivalence.

• The signature kernel score,

$$S_{\mathrm{Sig}}(F; y) = \mathbb{E}_{X, X' \sim F}[ k(X, X') ] - 2\mathbb{E}_{X \sim F}[ k(X, y) ]$$

is a strictly proper scoring rule; the minimiser of its expected value is uniquely the true distribution, both in deterministic and probabilistic settings (Dodson et al., 21 Oct 2025, Issa et al., 2023).

• For stochastic process discrimination:

$$d_k^2(\mu, \nu) = \mathbb{E}[k(X, X')] + \mathbb{E}[k(Y, Y')] - 2\mathbb{E}[k(X, Y)]$$

yields a consistent, nonparametric two-sample test for path laws (Chevyrev et al., 2018).

4. Applications: Probabilistic Modeling, Learning, and Testing

Signature kernels provide a theoretically sound and practically scalable basis for learning and model evaluation:

• Forecast Verification and Training: The signature kernel scoring rule is used in spatio-temporal probabilistic forecasting, neural weather models, and generative SDEs, offering higher discriminative power and strict propriety compared to CRPS or MSE (Dodson et al., 21 Oct 2025, Issa et al., 2023, Lu et al., 29 Jul 2024).
• Anomaly Detection: Embedded in one-class SVDD with manifold regularisation for time series anomaly detection, the kernel enables tight generalization bounds and efficient representer theorem solutions (Arashloo, 31 Jul 2025).
• Nonparametric Hypothesis Testing: Characteristicness underpins consistent Maximum Mean Discrepancy and conditional independence tests in causal discovery for stochastic processes (Chevyrev et al., 2018, Manten et al., 28 Feb 2024).
• Sparse Feature Extraction: By constructing signature-space filters, single or specific groups of signature coefficients can be isolated for sparse control (e.g., in high-dimensional CDEs) (Shmelev et al., 11 Dec 2024).
• Path-dependent PDE Learning: The RKHS defined by the signature kernel enables provably convergent mesh-free solvers for path-dependent PDEs, e.g., in rough volatility pricing (Pannier et al., 18 Mar 2024).
• Tree and Structured Data: Extensions naturally address hierarchical (streaming trees) and multi-modal sequential data (Cochrane et al., 2021).

5. Numerical Considerations, Scalability, and Software

Practical signature kernel computation is governed by several critical factors:

• PDE-Based Solvers: Explicit finite-difference and higher-order schemes for the Goursat PDE enable $O(L^2d)$ complexity for sequence length $L$ and dimension $d$; GPU implementations reduce effective time to $O(Ld)$ given sufficient hardware (Salvi et al., 2020, Shmelev et al., 12 Sep 2025).
• Memory-Efficient Approaches: Local Neumann expansion (tilewise) with dynamic truncation achieves drastic memory reduction, scaling to sequences with $L>10^5$ (Tamayo-Rios et al., 27 Feb 2025).
• Random Feature/Projection Methods: Random Fourier, diagonal-projection, and tensor-sketch signature features allow linear scaling in both dataset and sequence sizes, with provable uniform approximation guarantees (Toth et al., 2023, Tóth et al., 13 Jan 2025).
• Parameter Selection/Preprocessing:
  • Normalize variables to mean zero, variance one.
  • Always include basepoint and time augmentations to ensure injectivity.
  • RBF static