Computing Branched Log Signatures

Let \(\mathcal{H}\) be a Hopf algebra with counit \(\varepsilon\). For an element \(x \in \mathcal{H}\) the logarithm is defined by

\[\log_*(x) = \sum_{k \geq 1} \frac{(-1)^{k-1}}{k} (x-\varepsilon)^{*k}.\]

Applying this logarithm map to the branched signature yields the branched log signature. The logarithm is taken in the same Hopf algebra as the branched signature itself: the BCK algebra when planar=False and the MKW ordered forest algebra when planar=True.

Preparing for Branched Log Signature Computations

Before computing a branched log signature, pysiglib requires a call to pysiglib.prepare_branched_sig. This will pre-compute and cache the tree or ordered-forest basis and the coproduct tables required for the computation. This function should be run only once before the computation, for each required (dimension, degree, planar) combination.

import numpy as np
import pysiglib

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

for i in range(10):
    X = np.random.rand(200, 2)

    X_bck_logsig = pysiglib.branched_log_sig(X, 3)
    X_mkw_logsig = pysiglib.branched_log_sig(X, 3, planar=True)

The prepared object depends on the value of planar. Preparing with planar=False is not sufficient for planar=True, and conversely.

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}
}