Computing Branched Signatures

Branched signatures are a tree-indexed variant of path signatures. Instead of recording iterated integrals indexed by words, they record coefficients indexed by rooted trees. These coefficients are the natural coordinates for branched rough paths, B-series, and their planar Lie-Butcher analogues.

Non-Planar and Planar Branched Signatures

pysiglib currently supports two branched-signature algebras.

With planar=False (the default), the basis is made of decorated non-planar rooted trees and the coproduct is the Butcher-Connes-Kreimer (BCK) coproduct. This is the usual branched rough path setting over Euclidean space.

With planar=True, the basis is made of decorated ordered forests of planar rooted trees and the coproduct is the Munthe-Kaas-Wright (MKW) coproduct. This is the algebra used for planar branched rough paths and Lie-Butcher series.

The following tables compare the sizes of signatures and branched signatures with scalar_term=False.

Signature

Degree	d=1	d=2	d=3
1	1	2	3
2	2	6	12
3	3	14	39
4	4	30	120
5	5	62	363

Non-Planar Branched Signature

Degree	d=1	d=2	d=3
1	1	2	3
2	2	6	12
3	4	20	57
4	8	72	303
5	17	286	1788

Planar Branched Signature

Degree	d=1	d=2	d=3
1	1	2	3
2	3	10	21
3	8	50	156
4	22	274	1290
5	64	1618	11496

Preparing and indexing

Before computing branched signatures, call pysiglib.prepare_branched_sig once for each (dimension, degree, planar) combination that will be used. The preparation step builds the tree or ordered-forest basis and the coproduct tables used for convolution.

import numpy as np
import pysiglib

pysiglib.prepare_branched_sig(2, 3)
pysiglib.prepare_branched_sig(2, 3, planar=True)

path = np.array([[0.0, 0.0], [0.2, -0.3], [0.7, 0.4]])

# BCK non-planar branched signature.
bsig = pysiglib.branched_sig(path, 3)

# MKW planar branched signature.
planar_bsig = pysiglib.branched_sig(path, 3, planar=True)

Citation

If you found this library useful in your research, please consider citing the paper:

@article{shmelev2025pysiglib,
  title={pySigLib-Fast Signature-Based Computations on CPU and GPU},
  author={Shmelev, Daniil and Salvi, Cristopher},
  journal={arXiv preprint arXiv:2509.10613},
  year={2025}
}