pysiglib.sig_mmd

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pysiglib.sig_mmd#

Added in version v0.2.1.

sig_mmd(sample1, sample2, dyadic_order, *, static_kernel=None, time_aug=False, lead_lag=False, end_time=1.0, n_jobs=1, max_batch=-1)[source]#

Computes the squared maximum mean discrepancy (MMD)

\[d(\mu, \nu)^2 := \sup_f(\mathbb{E}_{x \sim \mu}[f(x)] - \mathbb{E}_{y \sim \nu}[f(y)]).\]
\[= \mathbb{E}_{xx' \sim \mu}[k(x,x')] - 2\mathbb{E}_{x,y \sim \mu \times \nu}[k(x,y)] + \mathbb{E}_{y,y' \sim \nu}[k(y,y')].\]

Given a batch of sample paths \(\{x_i\}_{i=1}^m \sim \mu\) and \(\{y_j\}_{j=1}^n \sim \nu\), the MMD is computed using the unbiased estimator

\[\widehat{d}(\mu, \nu)^2 = \frac{1}{m(m-1)}\sum_{j \neq i}k(x_i, x_j) - \frac{2}{mn}\sum_{i,j}k(x_i, x_j) + \frac{1}{n(n-1)}\sum_{j \neq i} k(y_i, y_j).\]

Optionally, a static kernel can be specified. For details, see the documentation on static kernels.

Parameters:

  • sample1 (numpy.ndarray | torch.tensor) – The batch of sample paths drawn from \(\mu\), of shape (*batch_shape, length_1, dimension). All dimensions before length_1 are batch dimensions and are flattened to a single batch \(m\) of paths.

  • sample2 (numpy.ndarray | torch.tensor) – The batch of sample paths drawn from \(\nu\), of shape (*batch_shape, length_2, dimension). All dimensions before length_2 are flattened to a single batch \(n\) of paths. Independent of sample1’s batch shape.

  • dyadic_order (int | tuple) – If set to a positive integer \(\lambda\), will refine the paths by a factor of \(2^\lambda\). If set to a tuple of positive integers \((\lambda_1, \lambda_2)\), will refine the first path by \(2^{\lambda_1}\) and the second path by \(2^{\lambda_2}\).

  • static_kernel (None | pysiglib.StaticKernel) – Static kernel passed to the signature kernel computation. If None (default), the linear kernel will be used. For details, see the documentation on static kernels.

  • time_aug (bool) – If set to True, will compute the signature of the time-augmented path, \(\hat{x}_t := (t, x_t)\), defined as the original path with an extra channel set to time, \(t\). This channel spans \([0, t_L]\), where \(t_L\) is given by the parameter end_time.

  • lead_lag (bool) – If set to True, will compute the signature of the path after applying the lead-lag transformation.

  • end_time (float) – End time for time-augmentation, \(t_L\).

  • n_jobs (int) – (Only applicable to CPU computation) Number of threads to run in parallel. If n_jobs = 1, the computation is run serially. If set to -1, all available threads are used. For n_jobs below -1, (max_threads + 1 + n_jobs) threads are used. For example if n_jobs = -2, all threads but one are used.

  • max_batch (int) – Maximum batch size to run in parallel. If the computation is failing due to insufficient memory, this parameter should be decreased. If set to -1, the entire batch is computed in parallel.

Returns:

Signature MMD

Return type:

numpy.ndarray | torch.tensor

Example:#

import torch
import pysiglib

sample1 = torch.rand((20, 100, 5))
sample2 = torch.rand((15, 100, 5))
mmd = pysiglib.sig_mmd(sample1, sample2, dyadic_order=2)
print(mmd)

# MMD with a static kernel and time augmentation
import torch
import pysiglib

sample1 = torch.rand((20, 100, 5))
sample2 = torch.rand((15, 100, 5))
rbf = pysiglib.RBFKernel(sigma=0.5)
mmd = pysiglib.sig_mmd(
    sample1, sample2,
    dyadic_order=2,
    static_kernel=rbf,
    time_aug=True,
    max_batch=8,
)
print(mmd)

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}
}
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