Branched Signature Model

ArXiv ID: 2511.00018 “View on arXiv”

Authors: Munawar Ali, Qi Feng

Abstract

In this paper, we introduce the branched signature model, motivated by the branched rough path framework of [“Gubinelli, Journal of Differential Equations, 248(4), 2010”], which generalizes the classical geometric rough path. We establish a universal approximation theorem for the branched signature model and demonstrate that iterative compositions of lower-level signature maps can approximate higher-level signatures. Furthermore, building on the existence of the extension map proposed in [“Hairer-Kelly. Annales de l’Institue Henri Poincaré, Probabilités et Statistiques 51, no. 1 (2015)”], we show how to explicitly construct the extension of the original paths into higher-dimensional spaces via a map $Ψ$, so that the branched signature can be realized as the classical geometric signature of the extended path. This framework not only provides an efficient computational scheme for branched signatures but also opens new avenues for data-driven modeling and applications.

Keywords: rough path theory, universal approximation, signature transforms, branched signature, stochastic analysis, Mathematical Modeling

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is heavily theoretical, focusing on abstract mathematical structures like Hopf algebras and rough paths, with numerous advanced formulas and proofs. It lacks any mention of backtests, datasets, or empirical performance metrics, placing it firmly in the theoretical research quadrant.

  flowchart TD
    A["Research Goal"] --> B["Methodology"]
    B --> C["Data/Inputs"]
    C --> D{"Computational Processes"}
    D --> E["Key Findings/Outcomes"]
    
    A --> A1["Goal: Develop branched signature model<br>for universal approximation and data modeling"]
    
    B --> B1["1. Establish universal approximation theorem"]
    B --> B2["2. Show iterative composition of lower-level<br>signatures approximates higher-level signatures"]
    B --> B3["3. Construct explicit path extension map Ψ<br>(building on Hairer-Kelly extension theorem)"]
    
    C --> C1["Paths from branched rough path framework<br>(Gubinelli, 2010)"]
    C --> C2["Extended paths in higher-dimensional space<br>(via map Ψ)"]
    
    D --> D1["Compute branched signatures as<br>classical geometric signatures of extended paths"]
    D --> D2["Efficient computational scheme for<br>branched signature computation"]
    
    E --> E1["Universal approximation capability proven"]
    E --> E2["Branched signatures realized as<br>classical geometric signatures of extended paths"]
    E --> E3["New avenue for data-driven modeling and applications"]
    
    style A fill:#e1f5fe
    style B fill:#f3e5f5
    style C fill:#e8f5e8
    style D fill:#fff3e0
    style E fill:#fce4ec