2026/07/28 by Emilio Ferrucci, Oliver Perrée, Terry Lyons
Computer Science · Mathematics · #cs.NA #math.NA #math.PR #msc:34A30 #msc:60L10 #msc:60L20 #msc:60L90 #msc:65L70 #msc:68P20
arxiv created 2026/07/28 · arxiv updated 2026/07/30
Paths X \colon [0,T] → \mathbb Rd are traditionally stored in finite memory as time series. Recent research has underscored the benefits of instead representing them as collections of iterated integrals \∫0 < u1 < … < un < T d Xu1 ⊗ ⋯ ⊗ d Xun\n = 0N. These two encodings can be viewed as the extrema on a two-parameter spectrum of representations of the path as degree-N signatures on m intervals in a partition of [0,T]. We ask the question of which such representation takes up the least amount of memory, measured as number of real values needed to store the truncated log-signature, subject to the constraint of it being able to approximate solutions to linear controlled differential equations (CDEs) d Y = AY d X with |A| ≤ r at accuracy at least ε. Estimating the error in terms of the length of X, we find that the optimal representation generally lies strictly in between the two naive choices N = 1 or m = 1, and derive its asymptotics as r → ∞ and ε → 0+. Similar considerations can be made when estimating the error in terms of the p-variation norm of X: in this regime we prove an error bound of the degree-N Euler scheme for linear CDEs with decay in both m and (factorially) in N with the other arbitrarily fixed. We conclude by setting up the analogous problem for SDEs, with the error measured in L2, and derive a similar L2-Euler error estimate for Itô SDEs with drift. We include an empirical study of the optimisation problem, which we demonstrate for toy examples of p-rough paths and for fractional Brownian motion.