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Free probability, path developments and signature kernels as universal scaling limits

Thomas Cass, William F. Turner

Abstract

Random developments of a path into a matrix Lie group $G_N$ have recently been used to construct signature-based kernels on path space. Two examples include developments into GL$(N;\mathbb{R})$ and $U(N;\mathbb{C})$, the general linear and unitary groups of dimension $N$. For the former, [MLS23] showed that the signature kernel is obtained via a scaling limit of developments with Gaussian vector fields. The second instance was used in [LLN23] to construct a metric between probability measures on path space. We present a unified treatment to obtaining large $N$ limits by leveraging the tools of free probability theory. An important conclusion is that the limiting kernels, while dependent on the choice of Lie group, are nonetheless universal limits with respect to how the development map is randomised. For unitary developments, the limiting kernel is given by the contraction of a signature against the monomials of freely independent semicircular random variables. Using the Schwinger-Dyson equations, we show that this kernel can be obtained by solving a novel quadratic functional equation. We provide a convergent numerical scheme for this equation, together with rates, which does not require computation of signatures themselves.

Free probability, path developments and signature kernels as universal scaling limits

Abstract

Random developments of a path into a matrix Lie group have recently been used to construct signature-based kernels on path space. Two examples include developments into GL and , the general linear and unitary groups of dimension . For the former, [MLS23] showed that the signature kernel is obtained via a scaling limit of developments with Gaussian vector fields. The second instance was used in [LLN23] to construct a metric between probability measures on path space. We present a unified treatment to obtaining large limits by leveraging the tools of free probability theory. An important conclusion is that the limiting kernels, while dependent on the choice of Lie group, are nonetheless universal limits with respect to how the development map is randomised. For unitary developments, the limiting kernel is given by the contraction of a signature against the monomials of freely independent semicircular random variables. Using the Schwinger-Dyson equations, we show that this kernel can be obtained by solving a novel quadratic functional equation. We provide a convergent numerical scheme for this equation, together with rates, which does not require computation of signatures themselves.
Paper Structure (15 sections, 11 theorems, 113 equations, 3 figures)

This paper contains 15 sections, 11 theorems, 113 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Letters, Words and Graphs
  4. The Signature
  5. The Signature Kernel is a Universal Limit
  6. Universal Limit of Unitary Developments and a Functional Equation
  7. Non-Crossing Partitions, Dyck Words and Trees
  8. Generations
  9. A Limiting Functional Equation
  10. Numerical Schemes
  11. Explicit Scheme
  12. Implicit Scheme
  13. Randomised CDEs
  14. Comparing Numerical Solvers
  15. Proof of Convergence of Numerical Scheme

Key Result

Theorem 2.1

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space supporting for each $N\in\mathbb{N}$ and $1\leq i \leq d$, a random matrix $A_i^N:\Omega \to \mathfrak{g}_N$. Assume that for all $k\in\mathbb{N}$, and that $\mathbb{E}[A_i^N(m,l)]=0$ and $\mathbb{E}[\vert A_i^N(m,l)\vert^2]=1$. Suppose also that there exists some $\kappa>0$ such that for every $p\in\mathbb{N}$ where $\Gamma(\cdot)$ i

Figures (3)

  • Figure 1: Convergence for the implicit numerical scheme $K_{\gamma^{\pi(\lambda)}}^r$ and the randomised scheme $\text{r}K_{\gamma}^{M,N}$ for $d=1$ and $d=20$. For each dimension, we generate two examples of a piecewise-linear interpolation of fractional Brownian motion with Hurst parameter $H=0.75$ on a uniform partition $\pi$ of $[0,1]$ of size $15$. The blue line corresponds to the lower $\mathrm{x}-$axis, indexed by the dyadic order, and the red line corresponds to the upper $\mathrm{x}-$axis, indexed by the dimension of the matrices $A_i^{M,N}$; the number of Monte-Carlo samples is fixed at $M=50$. The dotted black line is the true value $K_\gamma$, computed in $d=1$ by \ref{['rem: exact_sol']} and by $\text{r}K_{\gamma}^{125, 450}$ in $d=20$.
  • Figure 2: A comparison of the computational time scaling for the implicit numerical scheme and the randomised scheme when path length and path dimension are varied. The hyper-parameters of $\text{r}K_{\gamma}^{M,N}$ are fixed at $N=20$ and $M=10$.
  • Figure 3: Computational time vs. N for a path with $128$ time steps in $d=10$. The remaining hyper-parameter of $\text{r}K_{\gamma}^{M,\cdot}$ is fixed at $M=2$.

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.4
  • proof
  • proof : Proof of \ref{['thm: main_thm']}
  • Theorem 2.5
  • proof
  • ...and 28 more