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PreprintarXiv licencearXiv

Universal Varieties for Signatures of Membranes

  • Carlos Améndola
  • Gabriel Riffo
  • Leonard Schmitz
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Abstract

Membranes are sufficiently smooth vector-valued functions defined on the unit square, widely used in applications such as interpolation of image data. The signature of a membrane is a collection of nonlinear features that capture its essential geometry. We index these features in the graded Hopf algebra of two-parameter shuffle products and study the algebraic varieties for each graded component. At each truncation level, we obtain the dimension of this variety via a novel Witt-type formula. We propose a Chen-Chow theorem for membranes, provably at the second truncation level, and conjecturally for higher levels with extensive computational evidence in the computer algebra system OSCAR.

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Cite

BibTeXRIS
BibTeX
@misc{amendola2026universal,
  title = {{Universal Varieties for Signatures of Membranes}},
  author = {Carlos Améndola and Gabriel Riffo and Leonard Schmitz},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16724},
  eprint = {2610.16724},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16724},
}
RIS
TY  - UNPB
TI  - Universal Varieties for Signatures of Membranes
AU  - Carlos Améndola
AU  - Gabriel Riffo
AU  - Leonard Schmitz
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16724
UR  - https://arxiv.org/abs/2610.16724
AB  - Membranes are sufficiently smooth vector-valued functions defined on the unit square, widely used in applications such as interpolation of image data. The signature of a membrane is a collection of nonlinear features that capture its essential geometry. We index these features in the graded Hopf algebra of two-parameter shuffle products and study the algebraic varieties for each graded component. At each truncation level, we obtain the dimension of this variety via a novel Witt-type formula. We propose a Chen-Chow theorem for membranes, provably at the second truncation level, and conjecturally for higher levels with extensive computational evidence in the computer algebra system OSCAR.
ER  -