[论文解读] Signature Varieties of Splines

Carlos Améndola, Felix Lotter+1 more|arXiv (Cornell University)|Feb 13, 2026

Advanced Numerical Analysis Techniques被引用 0

一句话总结

论文引入并研究分段多项式路径（样条）的签名流形，通过代数参数化、字典和核心张量描述其结构，并在若干情形下确定维数与次数。它还分析签名映射的纤维以用于路径重建和学习应用。

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.

研究动机与目标

• Motivate the study of path signatures for splines and their algebraic-geometric properties.
• Define geometric and parametric spline signature varieties and analyze their basic algebraic structure.
• Develop dictionaries and core tensors to describe spline families via matrix-tensor actions.
• Compute dimensions and degrees for spline signature varieties in key cases.
• Investigate learning aspects and fiber structure of the signature map for spline reconstruction.

提出的方法

• Define truncated signature for piecewise polynomial splines and their regularity notions (geometric vs. parametric).
• Introduce signature varieties as Zariski closures of images of polynomial maps from spline spaces.
• Use dictionary/core-tensor framework to describe signatures via matrix-tensor congruence (A*C).
• Construct algebraically parametrized dictionaries (B_rho) for geometric splines and derive core tensors.
• Compute dimensions and degrees of spline varieties, including matrix-variety identifications and explicit formulas.

实验结果

研究问题

• RQ1 What are the algebraic properties (dimension, degree, equations) of the signature tensors of spline families?
• RQ2 How can one describe spline signatures via dictionaries and core tensors, and what does this imply for reconstruction/learning from signatures?
• RQ3 What is the behavior of signature varieties under regularity constraints and compositions of spline segments?
• RQ4 How do the fibers of the signature map look for spline classes, and can splines be reconstructed from their signature tensors?

主要发现

• Signature varieties of geometric and parametric splines are irreducible and sit inside the Lie-group–tensor framework; at large enough composition length they fill the ambient variety.
• For piecewise polynomial splines with regularity r, the matrix varieties S^r_{d,2,m} and P^r_{d,2,m} coincide with the matrix-variety M_{d,M-(l-1)r} when M and l satisfy the stated conditions.
• The dimension of the matrix variety M_{d,M-(l-1)r} is Md-(l-1)dr - binom{M-(l-1)r}{2} (Corollary 5.3).
• If M is odd, the degree of M_{d,M} is given by deg(M_{d,M}) = 2^{M-1} * Product_{n=0}^{d-M-1} binom(d+n}{d-M-n} / binom(2n+1}{n} (Theorem 5.4).
• For k>2, the paper uses computational methods to determine ideals of spline signature varieties, illustrating the approach with explicit examples (Example 5.6).
• Dictionaries and core tensors provide a constructive description: S^0_{d,k,m} is described by the orbit A*C with core C = sigma^{≤k}(PwMom^m) (Theorem 4.4).