Recovering the Length of a Path from its Signature

Abstract

We prove the length conjecture of J. Chang, T. Lyons and H. Ni for paths of bounded variation for arbitrary real Banach spaces and any reasonable tensor (algebra) norms. More precisely, we establish the missing lower bound and in this way show that the length of a tree-reduced path is recovered from the asymptotic behaviour of its normalised signature. The main technical argument uses the norm duality and shuffle algebra to express natural powers of polygonal length approximations as combinations of linear functionals acting on restricted signatures, which are then glued by suitable integrations into a linear functional of controlled growth acting on the full signature. In particular, the full proof combines tools of classical analysis and functional analysis, such as localisation by continuous cutoff functions, Stone-Weierstrass approximation of continuous functions by signature polynomials, the Hahn-Banach theorem and a support detection lemma for compactly supported real signed Borel measures.