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Signature Varieties of Splines

Carlos Améndola, Felix Lotter, Leonard Schmitz

TL;DR

This work develops an algebraic-geometry framework for the truncated signature tensors of piecewise polynomial paths (splines). By defining spline signature varieties as Zariski closures of signature maps and building algebraically parametrized dictionaries and core tensors, it connects spline structure with matrix-tensor congruence and provides explicit dimension and degree results, alongside algorithmic approaches for higher tensors. The paper combines symbolic methods (Macaulay2, OSCAR) and numerical continuation (HomotopyContinuation.jl) to compute invariants and study the learning problem via fibers of the signature map, including concrete recovery guarantees in the plane and extensive computational tables for more complex cases. The results illuminate how spline subclasses constrain signature varieties, reveal when learning from signatures is identifiable, and offer practical guidance for reconstructing splines from signature data in learning tasks. Overall, this work lays an explicit algebraic foundation for understanding and exploiting spline signatures in learning and geometry-informed analysis.

Abstract

Splines are central objects for the interpolation of discrete data via piecewise smooth paths. Their iterated-integral signature is an infinite collection of tensors which characterizes paths almost uniquely. We study truncations of this collection, which define algebraic maps from parameter space to tensor space. We prove that the images of these maps are given by orbits of a matrix-tensor action. Furthermore, taking the Zariski closure, we define and study varieties of spline signature tensors. We determine dimension and degree of these tensor varieties in a number of examples, relying on symbolic computations. With a view towards learning, constructing paths with a given signature tensor translates to studying the fibers of the signature map. We use computational methods to determine their cardinality, with a focus on its dependence on different classes of splines. We observe in explicit examples that reconstructing splines from a given signature tensor of a path yields close approximations of the original path.

Signature Varieties of Splines

TL;DR

This work develops an algebraic-geometry framework for the truncated signature tensors of piecewise polynomial paths (splines). By defining spline signature varieties as Zariski closures of signature maps and building algebraically parametrized dictionaries and core tensors, it connects spline structure with matrix-tensor congruence and provides explicit dimension and degree results, alongside algorithmic approaches for higher tensors. The paper combines symbolic methods (Macaulay2, OSCAR) and numerical continuation (HomotopyContinuation.jl) to compute invariants and study the learning problem via fibers of the signature map, including concrete recovery guarantees in the plane and extensive computational tables for more complex cases. The results illuminate how spline subclasses constrain signature varieties, reveal when learning from signatures is identifiable, and offer practical guidance for reconstructing splines from signature data in learning tasks. Overall, this work lays an explicit algebraic foundation for understanding and exploiting spline signatures in learning and geometry-informed analysis.

Abstract

Splines are central objects for the interpolation of discrete data via piecewise smooth paths. Their iterated-integral signature is an infinite collection of tensors which characterizes paths almost uniquely. We study truncations of this collection, which define algebraic maps from parameter space to tensor space. We prove that the images of these maps are given by orbits of a matrix-tensor action. Furthermore, taking the Zariski closure, we define and study varieties of spline signature tensors. We determine dimension and degree of these tensor varieties in a number of examples, relying on symbolic computations. With a view towards learning, constructing paths with a given signature tensor translates to studying the fibers of the signature map. We use computational methods to determine their cardinality, with a focus on its dependence on different classes of splines. We observe in explicit examples that reconstructing splines from a given signature tensor of a path yields close approximations of the original path.
Paper Structure (8 sections, 15 theorems, 64 equations, 5 figures, 6 tables)

This paper contains 8 sections, 15 theorems, 64 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Splines and their signature tensors
  3. Signature varieties
  4. Dictionaries and core tensors
  5. Dimension and degree of spline varieties
  6. Matrix varieties
  7. Higher tensor varieties
  8. Spline recovery

Key Result

Proposition 2.3

The signature of any piecewise polynomial path $X=X^{[1]} \sqcup \ldots \sqcup X^{[\ell]}$ is algebraic in the coefficients of the polynomials defining $X^{[1]},\ldots,X^{[\ell]}$.

Figures (5)

  • Figure 1: A geometric $(2,2)$-spline of regularity $1$
  • Figure 2: Images of $\mathsf{PwMom}^{m}$ for $m\in\mathrm{Comp}(3)$
  • Figure 3: Dictionaries for $\mathcal{P}^{1}_{2,\leq k,m}$
  • Figure 4: Two paths witnessing $\mathsf{PRdeg}(\mathcal{S}^r_{2,\leq 3,(2,1)}) = 2$.
  • Figure 5: Splines of regularity $1$ for different $m$, with the same signature up to level $4$. Control points are marked in red.

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4: Geometric and parametric splines
  • Remark 2.5
  • Example 2.6
  • Example 2.7
  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • ...and 41 more