log_signatures_pytorch

Differentiable log-signature and signature kernels implemented in PyTorch with both CPU-friendly and GPU-parallel execution paths.

What you'll find

• Batched signature and log-signature computation for tensors shaped (batch, length, dim) with optional streaming outputs at every step. For a single path, add a leading dimension via unsqueeze(0).
• Sliding-window signatures and log-signatures that reuse streamed prefixes (Chen identity) instead of re-computing each window independently.
• Hall-basis utilities (hall_basis, logsigdim, logsigkeys) plus Lyndon “words” helpers (lyndon_words, logsigdim_words, logsigkeys_words) for inspecting dimensions and basis labels.
• Two log-signature coordinate systems: Signatory-style “words” (Lyndon, default) and Hall.
• Two log-signature backends: the default signature→log path, and an incremental sparse BCH implementation for depths up to 4 (falls back otherwise).
• The implementation of signatures is structured after keras_sig, but only focuses on pytorch.
• Dependencies are kept minimal.

Installation

Requires Python 3.13+ and PyTorch ≥ 2.9 (CPU or CUDA builds work). To install from pypi using pip:

pip install log-signatures-pytorch

From the repository root:

uv venv
source .venv/bin/activate
uv sync                    # installs runtime deps + project in editable mode

Verify PyTorch is available:

python - <<'PY'
import torch
print("torch version:", torch.__version__)
print("cuda available:", torch.cuda.is_available())
PY

Quick start

Signature and log-signature of a single path

import torch
from log_signatures_pytorch import signature, log_signature, logsigdim_words

path = torch.tensor([[0.0, 0.0], [1.0, 1.0], [2.0, 0.0]]).unsqueeze(0)

sig = signature(path, depth=2)
print(sig.shape)           # torch.Size([1, 6]) = sum(width**k for k in 1..depth)

log_sig = log_signature(path, depth=2)
print(log_sig.shape)       # torch.Size([1, 3]) = logsigdim_words(2, 2)
print("logsigdim_words:", logsigdim_words(2, 2))  # 3

# Lyndon words coordinates (Signatory-style)
log_sig_words = log_signature(path, depth=2, mode="words")
print(log_sig_words.shape)  # torch.Size([1, 3])

Batched computation and streaming outputs

batch_paths = torch.tensor([
    [[0.0, 0.0], [1.0, 1.0]],
    [[0.0, 0.0], [2.0, 2.0]],
])

sig = signature(batch_paths, depth=2)
print(sig.shape)                 # torch.Size([2, 6])

log_sig_stream = log_signature(batch_paths, depth=2, stream=True)
print(log_sig_stream.shape)      # torch.Size([2, 1, 3]) -> (batch, steps, logsigdim)

# Streaming for a single path returns one row per increment (batch=1)
sig_stream = signature(path, depth=2, stream=True)
print(sig_stream.shape)          # torch.Size([1, 2, 6])

Sliding-window signatures and log-signatures

import torch
from log_signatures_pytorch import windowed_signature, windowed_log_signature

path = torch.tensor([[0.0, 0.0], [1.0, 1.0], [2.0, 0.0], [3.0, -1.0]]).unsqueeze(0)
width = path.shape[-1]
window_size = 4
hop_size = 2

win_sig = windowed_signature(path, depth=2, window_size=window_size, hop_size=hop_size)
print(win_sig.shape)   # torch.Size([batch, num_windows, 6])

win_logsig = windowed_log_signature(
    path, depth=2, window_size=window_size, hop_size=hop_size, mode="words"
)
print(win_logsig.shape)  # torch.Size([batch, num_windows, logsigdim_words(width, 2)])

Hall basis helpers

from log_signatures_pytorch import hall_basis, logsigkeys

basis = hall_basis(width=2, depth=2)
print(basis)          # [1, 2, (1, 2)]

keys = logsigkeys(width=2, depth=2)
print(keys)           # ['1', '2', '[1,2]'] (matches esig format)

# Lyndon words helpers
from log_signatures_pytorch import lyndon_words, logsigkeys_words
words = lyndon_words(width=2, depth=3)
print(words)          # [(1,), (2,), (1, 2), (1, 1, 2), (1, 2, 2)]
print(logsigkeys_words(width=2, depth=3))

Choosing computation mode

• method: log_signature(..., method="bch_sparse") uses the incremental BCH routine for depths supported by HallBCH (depth ≤ 4); otherwise it falls back to the default path.
• mode: log_signature(..., mode="words"|"hall") chooses the coordinate basis. Default is "words". BCH currently requires mode="hall"; the default path supports both.

Signature outputs exclude the empty word (dimension is sum(width**k for k=1..depth)); use logsigdim(width, depth) to size log-signature outputs.

GPU compile recommendations

• For fixed shapes with many repeated calls, torch.compile with mode reduce-overhead gives the fastest runtime for signature (~0.08–0.12 ms in our sweeps) and for log_signature (similarly sized speedups). First-call compile time can be large—especially for log-signatures—so cache per shape if you need to reuse compiled artifacts.
• For workloads with many varying shapes or when compile latency matters more than per-call speed, prefer none (no compile) or the lighter default mode. default is slower than reduce-overhead at runtime but compiles much faster; this tradeoff is more pronounced for log-signatures.
• The benchmark helper benchmarks/benchmark_batch_signature.py supports --target signature|log_signature, --compile-modes none reduce-overhead default, --measure-compile-time, and per-shape compile caching. CSVs land under benchmarks/results/ by default.

Testing and verification

• Run the suite: pytest tests -q
• Mathematical property checks are documented in tests/mathematical_verification_guide.md.

Documentation

Documentation is built using MkDocs with mkdocstrings and Material theme. To build and serve the documentation:

# Build static site
uv run mkdocs build

# Serve locally (with auto-reload on changes)
uv run mkdocs serve

The documentation will be available at http://127.0.0.1:8000 when serving locally.

References

• esig: https://github.com/datasig-ac-uk/esig

Kidger, Patrick and Lyons, Terry, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, ICLR 2021. Repository: https://github.com/patrick-kidger/signatory

Genet, R., & Inzirillo, H. (2025). Keras sig: Efficient path signature computation on gpu in keras 3. arXiv preprint arXiv:2501.08455. Repository: https://github.com/remigenet/keras_sig

License

MIT

Citation

If you use this software in your research or in your project, please cite it as follows:

BibTeX

@software{log_signatures_pytorch,
  author = {Aune, Erlend},
  title = {log-signatures-pytorch: Differentiable log-signature and signature kernels in PyTorch},
  version = {0.1.x},
  url = {https://github.com/froskekongen/log_signatures_pytorch},
  year = {2025},
  license = {MIT}
}

Plain text

Aune, Erlend. (2025). log-signatures-pytorch: Differentiable log-signature and signature kernels in PyTorch (Version 0.1.x). [Computer software]: https://github.com/froskekongen/log_signatures_pytorch