# Geometric Loss functions between sampled measures, images and volumes

The **GeomLoss** library provides efficient GPU implementations for:

Kernel norms (also known as Maximum Mean Discrepancies).

Hausdorff divergences, which are positive definite generalizations of the Chamfer-ICP loss and are analogous to

**log-likelihoods**of Gaussian Mixture Models.Debiased Sinkhorn divergences, which are affordable yet

**positive and definite**approximations of Optimal Transport (Wasserstein) distances.

It is hosted on GitHub
and distributed under the permissive MIT license.

GeomLoss functions
are available through the custom PyTorch layers
`SamplesLoss`

,
`ImagesLoss`

and
`VolumesLoss`

which allow you to work with weighted **point clouds** (of any dimension),
**density maps** and **volumetric segmentation masks**.
Geometric losses come with three backends each:

A simple

`tensorized`

implementation, for**small problems**(< 5,000 samples).A reference

`online`

implementation, with a**linear**(instead of quadratic)**memory footprint**, that can be used for finely sampled measures.A very fast

`multiscale`

code, which uses an**octree**-like structure for large-scale problems in dimension <= 3.

A typical sample of code looks like:

```
import torch
from geomloss import SamplesLoss # See also ImagesLoss, VolumesLoss
# Create some large point clouds in 3D
x = torch.randn(100000, 3, requires_grad=True).cuda()
y = torch.randn(200000, 3).cuda()
# Define a Sinkhorn (~Wasserstein) loss between sampled measures
loss = SamplesLoss(loss="sinkhorn", p=2, blur=.05)
L = loss(x, y) # By default, use constant weights = 1/number of samples
g_x, = torch.autograd.grad(L, [x]) # GeomLoss fully supports autograd!
```

GeomLoss is a simple interface for cutting-edge Optimal Transport algorithms. It provides:

Support for

**batchwise**computations.**Linear**(instead of quadratic)**memory footprint**for large problems, relying on the KeOps library for map-reduce operations on the GPU.Fast

**kernel truncation**for small bandwidths, using an octree-based structure.Log-domain stabilization of the Sinkhorn iterations, eliminating numeric

**overflows**for small values of \(\varepsilon\).Efficient computation of the

**gradients**, which bypasses the naive backpropagation algorithm.Support for unbalanced Optimal Transport, with a softening of the marginal constraints through a maximum

**reach**parameter.Support for the ε-scaling heuristic in the Sinkhorn loop, with kernel truncation in dimensions 1, 2 and 3. On typical 3D problems, our implementation is

**50-100 times faster**than the standard SoftAssign/Sinkhorn algorithm.

Note, however, that `SamplesLoss`

does *not* implement the
Fast Multipole
or Fast Gauss transforms.
If you are aware of a well-packaged implementation
of these algorithms on the GPU, please contact me!

The divergences implemented here are
all **symmetric**, **positive definite**
and therefore suitable for measure-fitting applications.
For positive input measures \(\alpha\) and \(\beta\),
our \(\text{Loss}\) functions are such that

**GeomLoss** can be used in a wide variety of settings,
from **shape analysis** (LDDMM, optimal transport…)
to **machine learning** (kernel methods, GANs…)
and **image processing**.
Details and examples are provided below:

# Licensing, academic use

This library is licensed under the permissive MIT license,
which is fully compatible with both **academic** and **commercial** applications.
If you use this code in a research paper, **please cite**:

```
@inproceedings{feydy2019interpolating,
title={Interpolating between Optimal Transport and MMD using Sinkhorn Divergences},
author={Feydy, Jean and S{\'e}journ{\'e}, Thibault and Vialard, Fran{\c{c}}ois-Xavier and Amari, Shun-ichi and Trouve, Alain and Peyr{\'e}, Gabriel},
booktitle={The 22nd International Conference on Artificial Intelligence and Statistics},
pages={2681--2690},
year={2019}
}
```