torch-kde 0.1.0

A differentiable implementation of kernel density estimation in PyTorch

TorchKDE :fire:

A differentiable implementation of kernel density estimation in PyTorch by Klaus-Rudolf Kladny.

$$\hat{f}(x) = \frac{1}{|H|^{\frac{1}{2}} n} \sum_{i=1}^n K \left( H^{-\frac{1}{2}} \left( x - x_i \right) \right)$$

Installation Instructions

Clone the repository, cd into the root directory and run

pip install .

Now you are ready to go! If you would also like to run the code in the Jupyter notebooks or contribute to this package, please also install the packages in the requirements.txt (inside of an environment):

pip install -r requirements.txt

What's included?

Kernel Density Estimation

The KernelDensity class supports the same operations as the KernelDensity class in scikit-learn, but implemented in PyTorch and differentiable with respect to input data. Here is a little taste:

from torchkde import KernelDensity
import torch

multivariate_normal = torch.distributions.MultivariateNormal(torch.ones(2), torch.eye(2))
X = multivariate_normal.sample((1000,)) # create data
X.requires_grad = True # enable differentiation
kde = KernelDensity(bandwidth=1.0, kernel='gaussian') # create kde object with isotropic bandwidth matrix
_ = kde.fit(X) # fit kde to data

X_new = multivariate_normal.sample((100,)) # create new data 
logprob = kde.score_samples(X_new)

logprob.grad_fn # is not None

You may also check out demo_kde.ipynb for a simple demo on the Bart Simpson distribution, which yields the following density estimate:

Tophat Kernel Approximation

The Tophat kernel is not differentiable at two points and has zero derivative everywhere else. Thus, we provide a differentiable approximation via a generalized Gaussian (see e.g. Pascal et al. for reference):

$$K^{\text{tophat}}(x; \beta) = \frac{\beta \Gamma \left( \frac{p}{2} \right) }{\pi^{\frac{p}{2}} \Gamma \left( \frac{p}{2\beta} \right) 2^{\frac{p}{2\beta}}} \text{exp} \left( - \frac{| x |_2^{2\beta}}{2} \right),$$

where $p$ is the dimensionality of $x$. Based on this kernel, we can approximate the Tophat kernel for large values of $\beta$, as shown in the following 1-dimensional example:

We note that for $\beta = 1$, this approximation corresponds to a Gaussian kernel. Also, while the approximation becomes better for large values of $\beta$, its gradients with respect to the input also become larger. This is a tradeoff that must be balanced when using this kernel.

Supported Settings

The current implementation provides the following functionality:

Feature	Supported Values
Kernels	Gaussian, Epanechnikov, Exponential, Tophat Approximation
Tree Algorithms	Standard
Bandwidths	Float (Isotropic bandwidth matrix), Scott, Silverman

Got an Extension? Create a Pull Request!

In case you do not know how to do that, here are the necessary steps:

1. Fork the repo
2. Create your feature branch (git checkout -b cool_tree_algorithm)
3. Run the unit tests (python -m tests.test_kde) and only proceed if the script outputs "OK".
4. Commit your changes (git commit -am 'Add cool tree algorithm')
5. Push to the branch (git push origin cool_tree_algorithm)
6. Open a Pull Request

Issues?

If you discover a bug or do not understand something, please create an issue or let me know directly at kkladny [at] tuebingen [dot] mpg [dot] de! I am also happy to take requests for implementing specific functionalities.

"In God we trust. All others must bring data."

— W. Edwards Deming