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Kernel Operations on the GPU, with autodiff, without memory overflows

The KeOps library lets you compute generic reductions of large 2d arrays whose entries are given by a mathematical formula. It combines a tiled reduction scheme with an automatic differentiation engine, and can be used through Matlab, NumPy or PyTorch backends. It is perfectly suited to the computation of Kernel dot products and the associated gradients, even when the full kernel matrix does not fit into the GPU memory.

Using the PyTorch backend, a typical sample of code looks like:

import torch
from pykeops.torch import Genred

# Kernel density estimator between point clouds in R^3
my_conv = Genred('Exp(-SqDist(x, y))',  # formula
                 ['x = Vi(3)',        # 1st input: dim-3 vector per line
                  'y = Vj(3)'],       # 2nd input: dim-3 vector per column
                 reduction_op='Sum',  # we also support LogSumExp, Min, etc.
                 axis=1)              # sum with respect to "j", result indexed by "i"

# Apply it to 2d arrays x and y with 3 columns and a (huge) number of lines
x = torch.randn(1000000, 3, requires_grad=True).cuda()
y = torch.randn(2000000, 3).cuda()
a = my_conv(x, y)  # shape (1000000, 1), a_i = sum_j exp(-|x_i-y_j|^2)
g_x = torch.autograd.grad((a ** 2).sum(), [x])  # KeOps supports autodiff!

KeOps allows you to leverage your GPU without compromising on usability. It provides:

  • Linear (instead of quadratic) memory footprint for Kernel operations.
  • Support for a wide range of mathematical formulas.
  • Seamless computation of derivatives, up to arbitrary orders.
  • Sum, LogSumExp, Min, Max but also ArgMin, ArgMax or K-min reductions.
  • A conjugate gradient solver for e.g. large-scale spline interpolation method (or kriging aka. Gaussian process regression).
  • An interface for block-sparse and coarse-to-fine strategies.
  • Support for multi GPU configurations.

KeOps can thus be used in a wide variety of settings, from shape analysis (LDDMM, optimal transport...) to machine learning (kernel methods, k-means...) or kriging (aka. Gaussian process regression). More details are provided below:

KeOps is licensed under the MIT license.

Projects using KeOps

As of today, KeOps provides core routines for:

  • Deformetrica, a shape analysis software developed by the Aramis Inria team.
  • GeomLoss, a multiscale implementation of Kernel and Wasserstein distances that scales up to millions of samples on modern hardware.
  • FshapesTk and the Shapes toolbox, two research-oriented LDDMM toolkits.


Feel free to contact us for any bug report or feature request:

Table of content

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.. toctree::
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   :caption: KeOps


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   :caption: PyKeops


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   :caption: KeopsLab


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   :caption: Keops++


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