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Discrete signature varieties

Carlo Bellingeri, Raul Penaguiao

TL;DR

This work builds an algebraic-geometry framework for discrete time-series invariants by studying the images of discrete signatures under weight-$h$ projections, forming the varieties $V_{d,h,N}$ and the universal $V_{d,h}$. It develops a dual Hopf-Lie algebraic backbone with shuffle/quasi-shuffle structures, Lyndon-basis decompositions, and associated Lie groups, enabling an affine-variety description and dimension formulas. A central contribution is the partial resolution of a Chen-Chow-type question in the complex setting, proving $hatG_{d,h}=V_{d,h}$ for $h=2$ via the reachability framework and providing explicit weight-two equations. The results unify combinatorial, algebraic, and geometric aspects of discrete signatures, and pave the way for a full Chen-Chow characterization and broader applications in time-series invariants and algebraic statistics.

Abstract

Discrete signatures are invariants computed from time series corresponding to the discretised version of the signature of paths. We study the algebraic varieties arising from their images, the discrete signature varieties. We introduce them and compute their dimension in many cases. From a particular subclass of these varieties, we derive a partial solution to the Chen-Chow theorem for complex-valued time series.

Discrete signature varieties

TL;DR

This work builds an algebraic-geometry framework for discrete time-series invariants by studying the images of discrete signatures under weight- projections, forming the varieties and the universal . It develops a dual Hopf-Lie algebraic backbone with shuffle/quasi-shuffle structures, Lyndon-basis decompositions, and associated Lie groups, enabling an affine-variety description and dimension formulas. A central contribution is the partial resolution of a Chen-Chow-type question in the complex setting, proving for via the reachability framework and providing explicit weight-two equations. The results unify combinatorial, algebraic, and geometric aspects of discrete signatures, and pave the way for a full Chen-Chow characterization and broader applications in time-series invariants and algebraic statistics.

Abstract

Discrete signatures are invariants computed from time series corresponding to the discretised version of the signature of paths. We study the algebraic varieties arising from their images, the discrete signature varieties. We introduce them and compute their dimension in many cases. From a particular subclass of these varieties, we derive a partial solution to the Chen-Chow theorem for complex-valued time series.
Paper Structure (4 sections, 18 theorems, 102 equations, 1 table)

This paper contains 4 sections, 18 theorems, 102 equations, 1 table.

Table of Contents

  1. Introduction
  2. Discrete signatures and their varieties
  3. Lie polynomials and Lie groups
  4. Universal varieties and Chen-Chow Theorem for discrete signatures

Key Result

Theorem 1.1

Fix $d, h$ integers $\geq 1$. The dimension of the variety $\hat{\mathcal{G}}_{d,h}$ is given by where $\mu$ is the Möbius function, $C(k)$ is the set of compositions of the integer $k$ and $\ell(\alpha)$ is the length of each composition.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1
  • Example 3.3: The Canonical Axis time series
  • Remark 3.4
  • Definition 3.5
  • Theorem 3.6
  • proof
  • Remark 3.7
  • Remark 3.8
  • ...and 39 more