Python Code for Kernel Method-based Triggering Kernel Estimator via Least Squares Loss

This library provides a kernel method-based triggering kernel estimator for a linear Hawkes process, implemented in Tensorflow. This method is based on a representer theorem that emerges under the principle of penalized least squares minimization. For details, see our ICLR2026 paper [1].

The code was tested on Python 3.10.8, tensorflow-deps 2.10.0, tensorflow-macos 2.10.0, and tensorflow-metal 0.6.0.

Installation

To install latest version:

pip install git+https://github.com/HidKim/K2Hawkes

Basic Usage

Import our tiggering kernel estimator class:

from HidKim_K2IE import k2_hawkes_rfm

Initialize our estimator:

k2h = k2_hawkes_rfm(kernel='gaussian', n_rand_feature=200, seed=0)

• kernel: string, default='gaussian'
  

  The kernel function: 'gaussian', 'laplace', and 'cauchy'.

• n_rand_feature: int, default=200
  

  The number of random Fourier features. Quasi-Monte Carlo method is applied to random Fourier feature generation.

• seed: int, default=0
  

  The seed for sampling Fourier features.

Fit our estimator with data:

time = k2h.fit(spk, T, gamma, b, support)

• spk: ndarray of shape (n_points, 2)
  

  The training time-point data: [time, dimension].
  e.g.) [ [0.1, 0], [0.2, 2], [0.6, 1] ] represents that events occurred at times 0.1, 0.2, and 0.6 in the 1st, the 3rd, and the 2nd dimensions of a Hawkes process, respectively.

• T: float
  

  The end of observation region [0, T].

• gamma: float
  

  The regularlization hyper-parameter '\gamma' in ICLR2026 paper.

• b: float
  

  The scale hyper-parameter for shift-invariant kernel function.
  e.g.) 'gaussian' kernel: k(t,t') = exp[-(b(t-t'))^2].

• support: float
  

  The support window for the triggering kernels.

• Return: float
  

  The execution time.

Predict triggering kernel on specified inputs:

trig_est = k2h.predict(x, edge)

• x: ndarray of shape (n_points,)
  

  The points on input space for evaluating triggering kernel values.

• edge: int, ndarray of shape (2,)
  

  The pair of dimensions that specifies the interaction direction. The 1st dimension is '0'.
  e.g.) [0, 2] represents the triggering kernel from the third dim to the 1st dim.

• Return: ndarray of shape (n_points,)
  

  The predicted values of the specified triggering kernel at the specified points.

Predict baseline intensities:

mu_est = k2h.get_mu()

• Return: ndarray of shape (n_dimensions,)
  

  The predicted values of baseline intensities.

Evaluate intensity functions on specified inputs under the estimated triggering kernels:

r_est = k2h.intensity(x, spk)

• x: ndarray of shape (n_points,)
  

  The time-points for evaluating intensity values.

• spk: ndarray of shape (n_points, 2)
  

  The event data observed during the period of interest for intensity estimation.

• Return: list of ndarray of shape (n_dimensions, (n_points,))
  

  The predicted values of intensity functions at the specified points.

Reference

1. Hideaki Kim, Tomoharu Iwata. "A Representer Theorem for Hawkes Processes via Penalized Least Squares Minimization", International Conference on Learning Representations, 2026.

@inproceedings{kim2026arepre,
  title={A Representer Theorem for Hawkes Processes via Penalized Least Squares Minimization},
  author={Kim, Hideaki and Iwata, Tomoharu},
  booktitle={International Conference on Learning Representations},
  year={2026}
}

License

Released under "SOFTWARE LICENSE AGREEMENT FOR EVALUATION". Be sure to read it.

Contact

Feel free to contact the author Hideaki Kim (hideaki.kin@ntt.com).