The high-level interface of KeOps is the LazyTensor and LazyTensor wrappers, which allows users to perform efficient, semi-symbolic computations on very large NumPy arrays or PyTorch tensors respectively. As displayed on this website’s front page, this new tensor type may be used with very little overhead:

# Create two arrays with 3 columns and a (huge) number of lines, on the GPU
import torch
x = torch.randn(1000000, 3, requires_grad=True).cuda()
y = torch.randn(2000000, 3).cuda()

# Turn our Tensors into KeOps symbolic variables:
from pykeops.torch import LazyTensor
x_i = LazyTensor( x[:,None,:] )  # x_i.shape = (1e6, 1, 3)
y_j = LazyTensor( y[None,:,:] )  # y_j.shape = ( 1, 2e6,3)

# We can now perform large-scale computations, without memory overflows:
D_ij = ((x_i - y_j)**2).sum(dim=2)  # Symbolic (1e6,2e6,1) matrix of squared distances
K_ij = (- D_ij).exp()               # Symbolic (1e6,2e6,1) Gaussian kernel matrix

# And come back to vanilla PyTorch Tensors or NumPy arrays using
# reduction operations such as .sum(), .logsumexp() or .argmin().
# Here, the kernel density estimation   a_i = sum_j exp(-|x_i-y_j|^2)
# is computed using a CUDA online map-reduce routine that has a linear
# memory footprint and outperforms standard PyTorch implementations
# by two orders of magnitude.
a_i = K_ij.sum(dim=1)  # Genuine torch.cuda.FloatTensor, a_i.shape = (1e6, 1),
g_x = torch.autograd.grad((a_i ** 2).sum(), [x])  # KeOps supports autograd!


Starting with the KeOps 101 tutorial, most examples in our gallery rely on LazyTensor or LazyTensors : going through this collection of real-life demos is probably the best way of getting familiar with the KeOps user interface.

Going further, please refer to the LazyTensor or LazyTensor API for an exhaustive list of all supported operations.