Higher Order Kernel Mean Embeddings for Stochastic Processes

This library provides estimators for the first and second order Maximum Mean Discrepancies (MMDs) with GPU support.

Installation

pip install git+https://github.com/maudl3116/higherOrderKME.git

Structure of the repository

• The higherOrderKME folder contains the implementation of the higher order MMDs (see sigkernel.py) and higher order distribution regression algorithms (see KES.py)
• The examples folder contains notebooks to perform two-sample tests, higher order distribution regression and causal graph discovery for stochastic processes.
• The data folder contains the data generators used for the experiments in the paper:
  • n_vcn_utils folder contains various utilies for the multibody interaction experiment (adapted from https://github.com/pairlab/v-cdn)
  • The option_utils folder contains various utilies for the optimal stopping time experiment (adapted from https://github.com/HeKrRuTe/OptStopRandNN)

How to use the library

import torch
from higherOrderKME import sigkernel

# Specify the static kernel (for linear kernel use sigkernel.LinearKernel())
static_kernel = sigkernel.RBFKernel(sigma=0.5)

# Specify dyadic order for PDE solver (int > 0, default 0, the higher the more accurate but slower)
dyadic_order = 2

# Specify the hyperparameter for the estimation of the conditional KME
lambda_ = 1e-5

# Initialize the corresponding signature kernel
signature_kernel = sigkernel.SigKernel(static_kernel, dyadic_order)

# Synthetic data
n, len_x, len_y, dim = 100, 10, 20, 2
x = torch.rand((n,len_x,dim), dtype=torch.float64, device='cuda') # shape (batch,len_x,dim)
y = torch.rand((n,len_y,dim), dtype=torch.float64, device='cuda') # shape (batch,len_y,dim)

# Compute the (classical) first order MMD distance between samples x ~ P and samples y ~ Q, where P,Q are two distributions on path space
mmd_order1 = signature_kernel.compute_mmd(x, y, order=1)

# Compute the second order MMD distance between samples x ~ P and samples y ~ Q, where P,Q are two distributions on path space
mmd_order2 = signature_kernel.compute_mmd(x, y, lambda_=lambda_, order=2)

Citation

@article{salvi2021higher,
  title={Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes},
  author={Salvi, Cristopher and Lemercier, Maud and Liu, Chong and Hovarth, Blanka and Damoulas, Theodoros and Lyons, Terry},
  journal={arXiv preprint arXiv:2109.03582},
  year={2021}
}