# TorchKDE
[](https://doi.org/10.5281/zenodo.14674657)
A differentiable implementation of [kernel density estimation](https://en.wikipedia.org/wiki/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
The torch-kde package can be installed via `pip`. Run
```bash
pip install torch-kde
```
Now you are ready to go! If you would also like to run the code from the Jupyter notebooks or contribute to this package, please also install the packages in the `requirements.txt`:
```bash
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](https://scikit-learn.org/dev/modules/generated/sklearn.neighbors.KernelDensity.html), but implemented in PyTorch and differentiable with respect to input data. Here is a little taste:
```python
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](https://www.stat.cmu.edu/~larry/=sml/densityestimation.pdf).
### 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.](https://arxiv.org/pdf/1302.6498) 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 $\beta$.
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:
<div align="center">
| Feature | Supported Values |
|--------------------------|-----------------------------|
| Kernels | Gaussian, Epanechnikov, Exponential, Tophat Approximation |
| Tree Algorithms | Standard |
| Bandwidths | Float (Isotropic bandwidth matrix), Scott, Silverman |
</div>
## 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.
<div align="center">
> "In God we trust. All others must bring data."
>
> — W. Edwards Deming
>
</div>
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