Scalable Signature Kernel Computations for Long Time Series via Local Neumann Series Expansions

TL;DR

PowerSig delivers a scalable, memory‑efficient method for computing the signature kernel of long, high‑dimensional time series by transforming the Goursat PDE into a Volterra integral equation and solving it via tilewise, adaptively truncated Neumann series. The approach partitions the domain into tiles, computes rapidly convergent local power series on each tile, and propagates boundary data along a directed tile graph, achieving substantial memory savings without sacrificing accuracy. Key contributions include a tile‑centered coefficient recursion, rigorous convergence via Neumann series, and a formal Gram‑matrix error bound, with empirical results showing superior accuracy and scalability on very long sequences and rough trajectories. The method maintains the usual $\mathcal{O}(\ell^2 d)$ time complexity but improves space to $\mathcal{O}(\ell P)$, enabling very long sequences (e.g., over a million points) on a single GPU and broad applicability to rough‑path–aided learning, finance, and signal processing.

Abstract

The signature kernel is a recent state-of-the-art tool for analyzing high-dimensional sequential data, valued for its theoretical guarantees and strong empirical performance. In this paper, we present a novel method for efficiently computing the signature kernel of long, high-dimensional time series via adaptively truncated recursive local power series expansions. Building on the characterization of the signature kernel as the solution of a Goursat PDE, our approach employs tilewise Neumann-series expansions to derive rapidly converging power series approximations of the signature kernel that are locally defined on subdomains and propagated iteratively across the entire domain of the Goursat solution by exploiting the geometry of the time series. Algorithmically, this involves solving a system of interdependent Goursat PDEs via adaptively truncated local power series expansions and recursive propagation of boundary conditions along a directed graph in a topological ordering. This method strikes an effective balance between computational cost and accuracy, achieving substantial performance improvements over state-of-the-art approaches for computing the signature kernel. It offers (a) adjustable and superior accuracy, even for time series with very high roughness; (b) drastically reduced memory requirements; and (c) scalability to efficiently handle very long time series (one million data points or more) on a single GPU. As demonstrated in our benchmarks, these advantages make our method particularly well-suited for rough-path-assisted machine learning, financial modeling, and signal processing applications involving very long and highly volatile sequential data.

Figures (10)

• Figure 1: Summary of our method (PowerSig) for computing the signature kernel of two time series via recursive local Neumann expansions. Panel A: The PDE $\frac{\partial^2 K}{\partial s \partial t}=\rho_{\bm{x},\bm{y}}K$ induces an $(\bm{x}, \bm{y})$-dependent partition of $[0,1]^2$ into tiles $T_{k,l}$. Arrows indicate the sequential propagation of boundary conditions across tiles, with decreasing colour intensity corresponding to later propagation steps. Tiles receiving arrows of the same colour form groups whose local series can be computed in parallel. Panel B: On each given tile $T_{k,l}$, the kernel admits the recursive local Neumann series expansion: $\kappa_{k,l}(s,t)=\sum_{n=0}^{\infty}T_{k,l}^n[\kappa_{k-1,l}(\sigma_k,\cdot)+\kappa_{k,l-1}(\cdot,\tau_l)-\kappa_{k-1,l-1}(\sigma_k,\tau_l)]=\sum_{i,j=1}^{\infty}c_{ij}(s-\sigma_k)^i(t-\tau_l)^j$, which converges uniformly on $(s,t)\in T_{k,l}$. These tilewise expansions depend on boundary values from neighbouring tiles ($\kappa_{k-1,l}$ and $\kappa_{k,l-1}$), with arrows indicating the directions of integration from boundaries to the tile interior. Panel C illustrates adaptive series truncation and final kernel assembly. Tile shading intensity encodes local truncation depth, which is adaptively determined by the magnitude of $\rho_{k,l}\equiv(\bm{x}_{k+1}-\bm{x}_k)(\bm{y}_{l+1}-\bm{y}_l)$. Darker tiles indicate the necessity for deeper (higher-order) expansions, while lighter tiles allow shallower truncation.
• Figure 2: Comparison of Mean Absolute Percentage Error (MAPE) between PowerSig and the PDE-based solver from KSig. Left: for two-dimensional Brownian motion paths on $[0,1]$ across increasing path lengths $\ell$. Right: for two-dimensional fractional Brownian motions of fixed length $\ell=51$ across increasingly irregular sample paths (decreasing Hurst index, swept through progressively rougher regimes); the right panel reports MAPE relative to the signature kernel truncated at level 180.
• Figure 3: Peak GPU memory usage (left) and runtime (right) for computing the signature kernel on two-dimensional Brownian motion paths, comparing PowerSig with the truncated-signature (KSig) and PDE-based (KSig PDE) solvers. PowerSig achieves substantially lower memory consumption and maintains computational feasibility for large $\ell$, well beyond the limits of the alternative methods.
• Figure 4: Bitcoin price regression (two-day rolling average). Top: training fit; bottom: test fit. PowerSig attains $2.81\%$ MAPE (under the default linear static kernel) compared to $3.23\%$ MAPE for the RBF-assisted KSig-PDE, while using only a fraction of the device memory.
• Figure 5: UEA Eigenworms classification across window lengths $L$. Test accuracy versus input window length $L$ for PowerSig, KSig-PDE, and the RFF baseline RFSF-TRP. PowerSig remains competitive and scales to $L=1048$ with $61.1\%$ accuracy; KSig-PDE is competitive up to $L=128$ before running out of memory (OOM). RFSF-TRP attains a slightly higher peak of $62.5\%$ at $L=128$ but OOMs for larger $L$, consistent with the storage advantages in Fig. \ref{['fig:memory_and_duration']}. This shows that extending the input window, enabled here at scale by PowerSig, can narrow performance gaps often ascribed to inductive bias while preserving feasibility.

Theorems & Definitions (19)

• Definition 2.1: Signature Kernel
• Proposition 2.2
• Lemma 2.3
• Lemma 2.4
• Proposition 2.5
• Example 2.6: Evaluating \ref{['eqn:recursion-on-tiles']} on $T_{1,1}$, $T_{1,2}$, and $T_{2,1}$
• Proposition 2.7
• Proposition 2.8
• proof : Proof of Lemma \ref{['lem:spectralradius']}
• proof