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The paper introduces Incremental Signature Contribution (ISC), a method that decomposes truncated path signatures into a temporally ordered sequence of elements, preserving the algebraic structure and expressivity of signatures while exposing temporal evolution. This allows for processing signature-based representations using sequential models, addressing the limitation of standard path signatures which collapse temporal structure. The authors then introduce ISC-Transformer (ISCT), an offline RL model integrating ISC into a standard Transformer, and demonstrate its effectiveness on benchmark tasks, particularly in settings requiring temporal sensitivity.
Path signatures, previously limited by their global view of time, can now be used in temporally sensitive control tasks thanks to a novel decomposition that exposes their internal temporal evolution.
Path signatures embed trajectories into tensor algebra and constitute a universal, non-parametric representation of paths; however, in the standard form, they collapse temporal structure into a single global object, which limits their suitability for decision-making problems that require step-wise reactivity. We propose the Incremental Signature Contribution (ISC) method, which decomposes truncated path signatures into a temporally ordered sequence of elements in the tensor-algebra space, corresponding to incremental contributions induced by last path increments. This reconstruction preserves the algebraic structure and expressivity of signatures, while making their internal temporal evolution explicit, enabling processing signature-based representations via sequential modeling approaches. In contrast to full signatures, ISC is inherently sensitive to instantaneous trajectory updates, which is critical for sensitive and stability-requiring control dynamics. Building on this representation, we introduce ISC-Transformer (ISCT), an offline reinforcement learning model that integrates ISC into a standard Transformer architecture without further architectural modification. We evaluate ISCT on HalfCheetah, Walker2d, Hopper, and Maze2d, including settings with delayed rewards and downgraded datasets. The results demonstrate that ISC method provides a theoretically grounded and practically effective alternative to path processing for temporally sensitive control tasks.