Linear independence properties of the signature components of time-augmented stochastic processes

TL;DR

The paper addresses linear independence and span properties of signature components for time-augmented stochastic processes. By introducing bases of words and leveraging shuffle algebra, Chen's relation, and pure-word decompositions, it characterizes subfamilies of signature terms that span the same space as all terms up to order $N$, irrespective of the underlying process law. It recovers known results (e.g., suffix/prefix-based bases) and demonstrates optimal computation times for these bases via forward/backward Chen recursions. For Itô SDEs with additive Brownian noise, the subfamilies corresponding to these bases remain linearly independent in $L^2$, and this independence persists under discretization with sufficiently small time steps. The numerical experiments corroborate the theoretical findings, showing comparable predictive performance with substantial reductions in feature dimension and training time, supporting suffix-based feature selection in signature regression tasks.

Abstract

The addition of the running time as a component of a path before computing its signature is a widespread approach to ensure the one-to-one property between them and leads to universal approximation theorems (Cuchiero, Primavera and Svaluto-Ferro, 2023). However, this also leads to the linear dependence of the components of the terminal value of the signature of the time-augmented path. More precisely, for a given natural number $N$, the signature components associated with words of length $N$ have the same linear span as the signature components associated with words of length not greater than $N$. We generalize this result by exhibiting other subfamilies of signature components with the same spanning properties. In particular we recover the result of Dupire and Tissot-Daguette which states that the spanning of the iterated integrals with the last integrator different from the time variable is the same as the spanning of all iterated integrals. We check that this choice leads to the minimal computation time when the terms of the signature are calculated using Chen's relation in a backward way. The same optimal computation time is symmetrically achieved in a forward way for the iterated integrals with the first integrator different from the time variable. Building on these results, we derive several results regarding the linear independence of the signature components of a time-augmented stochastic process. We show that if the stochastic process we consider is solution to some SDE with additive Brownian noise then any subfamily of components proposed previously is linearly independent. We also prove that the linear independence of these subfamilies of components is still true when we consider the discretization of the sample paths of this stochastic process on a grid with a sufficiently small discretization time step.

Figures (4)

• Figure 1: Computation time of signature components and comparison with theoretical scaling. We sample $200$ batches of size $n=500$ of discretized Brownian motion sample paths, and we compute the empirical mean over the batches of the time required to obtain the signature components.
• Figure 2: Computation time for $\hat{\beta}_{\text{all}}(\hat{\lambda}_{\text{all}})$ and $\hat{\beta}_{\text{suffix}}(\hat{\lambda}_{\text{suffix}})$. We sample $200$ batches of size $n=500$ of discretized Brownian motion paths, and we compute the empirical mean over the batches of the time required to obtain $\hat{\beta}_{\text{all}}(\hat{\lambda}_{\text{all}})$ (resp. $\hat{\beta}_{\text{suffix}}(\hat{\lambda}_{\text{suffix}})$) once the design matrix $\mathbf{X}_{\text{all}}$ (resp. $\mathbf{X}_{\text{suffix}}$) is computed. Note that the fitting time does not depend on the choice of $\beta_{\text{true}} \in \{(1,1,\cdots,1)^*, (1,2,\cdots,2^{11} - 1)^*, (2^{11}-1,\cdots,2,1)^*\}$ nor on whether $(X_t)_{t \in [0,T]}$ is a Brownian motion or an Ornstein-Uhlenbeck process.
• Figure 3: Difference of generalization errors: Brownian motion case.
• Figure 4: Difference of generalization errors: Ornstein-Uhlenbeck case.

Theorems & Definitions (64)

• Definition 1: Concatenation
• Definition 2: Shuffle product
• Example 1
• Remark 1
• Definition 3: Dual bracket
• Proposition 1: Shuffle product property
• Definition 4: Basis of words for $\mathcal{W}_N$
• Definition 5: Pure word
• Definition 6
• Definition 7: Basis of words for $\mathcal{W}_N({{\boldsymbol{\gamma}}})$