A wrapper for NumPy and PyTorch arrays

KeOps brings semi-symbolic calculus to modern computing libraries: it alleviates the need for huge intermediate variables such as kernel or distance matrices in machine learning and computational geometry.

First steps

A simple interface to the KeOps inner routines is provided by the pykeops.numpy.LazyTensor or pykeops.torch.LazyTensor symbolic wrapper, to be used with NumPy arrays or PyTorch tensors respectively.

To illustrate its main features on a simple example, let’s generate two point clouds \((x_i)_{i\in[1,M]}\) and \((y_j)_{j\in[1,N]}\) in the unit square:

import numpy as np

M, N = 1000, 2000
x = np.random.rand(M, 2)
y = np.random.rand(N, 2)

With NumPy, an efficient way of computing the index of the nearest y-neighbor

\[\sigma(i) = \arg \min_{j\in [1,N]} \| x_i - y_j\|^2\]

for all points \(x_i\) is to perform a numpy.argmin() reduction on the M-by-N matrix of squared distances

\[D_{i,j} = \|x_i-y_j\|^2,\]

computed using tensorized, broadcasted operators:

x_i = x[:, None, :]  # (M, 1, 2) numpy array
y_j = y[None, :, :]  # (1, N, 2) numpy array

D_ij = ((x_i - y_j) ** 2).sum(-1)  # (M, N) array of squared distances |x_i-y_j|^2
s_i = np.argmin(D_ij, axis=1)  # (M,)   array of integer indices
print(s_i[:10])

That’s good! Going further, we can speed-up these computations using the CUDA routines of the PyTorch library:

import torch

use_cuda = torch.cuda.is_available()
tensor = torch.cuda.FloatTensor if use_cuda else torch.FloatTensor

x_i = tensor(x[:, None, :])  # (M, 1, 2) torch tensor
y_j = tensor(y[None, :, :])  # (1, N, 2) torch tensor

D_ij = ((x_i - y_j) ** 2).sum(-1)  # (M, N) tensor of squared distances |x_i-y_j|^2
s_i = D_ij.argmin(dim=1)  # (M,)   tensor of integer indices
print(s_i[:10])

But can we scale to larger point clouds? Unfortunately, tensorized codes will throw an exception as soon as the M-by-N matrix \((D_{i,j})\) stops fitting contiguously on the device memory. This generally happens when \(\sqrt{MN}\) goes past a hardware-dependent threshold in the [5,000; 50,000] range:

M, N = (100000, 200000) if use_cuda else (1000, 2000)
x = np.random.rand(M, 2)
y = np.random.rand(N, 2)

x_i = tensor(x[:, None, :])  # (M, 1, 2) torch tensor
y_j = tensor(y[None, :, :])  # (1, N, 2) torch tensor

try:
    D_ij = ((x_i - y_j) ** 2).sum(-1)  # (M, N) tensor of squared distances |x_i-y_j|^2
except RuntimeError as err:
    print(err)

That’s unfortunate… And unexpected! After all, modern GPUs routinely handle the real-time rendering of scenes with millions of triangles moving around. So how do graphics programmers achieve such a level of performance?

The key to efficient numerical schemes is to remark that even though the distance matrix \((D_{i,j})\) is not sparse in the traditional sense, it definitely is compact from a computational perspective. Since its coefficients are fully described by two lists of points and a symbolic formula, sensible implementations should compute required values on-the-fly… and bypass, lazily, the cumbersome pre-computation and storage of all pairwise distances \(\|x_i-y_j\|^2\).

from pykeops.numpy import LazyTensor as LazyTensor_np

x_i = LazyTensor_np(
    x[:, None, :]
)  # (M, 1, 2) KeOps LazyTensor, wrapped around the numpy array x
y_j = LazyTensor_np(
    y[None, :, :]
)  # (1, N, 2) KeOps LazyTensor, wrapped around the numpy array y

D_ij = ((x_i - y_j) ** 2).sum(-1)  # **Symbolic** (M, N) matrix of squared distances
print(D_ij)

With KeOps, implementing lazy numerical schemes really is that simple! Our LazyTensor variables are encoded as a list of data arrays plus an arbitrary symbolic formula, written with a custom mathematical syntax that is modified after each “pythonic” operation such as -, **2 or .exp().

We can then perform a pykeops.torch.LazyTensor.argmin() reduction with an efficient Map-Reduce scheme, implemented as a templated CUDA kernel around our custom formula. As evidenced by our benchmarks, the KeOps routines have a linear memory footprint and generally outperform tensorized GPU implementations by two orders of magnitude.

s_i = D_ij.argmin(dim=1).ravel()  # genuine (M,) array of integer indices
print("s_i is now a {} of shape {}.".format(type(s_i), s_i.shape))
print(s_i[:10])

Going further, we can combine LazyTensors using a wide range of mathematical operations. For instance, with data arrays stored directly on the GPU, an exponential kernel dot product

\[a_i = \sum_{j=1}^N \exp(-\|x_i-y_j\|)\cdot b_j\]

in dimension D=10 can be performed with:

from pykeops.torch import LazyTensor

D = 10
x = torch.randn(M, D).type(tensor)  # M target points in dimension D, stored on the GPU
y = torch.randn(N, D).type(tensor)  # N source points in dimension D, stored on the GPU
b = torch.randn(N, 4).type(
    tensor
)  # N values of the 4D source signal, stored on the GPU

x.requires_grad = True  # In the next section, we'll compute gradients wrt. x!

x_i = LazyTensor(x[:, None, :])  # (M, 1, D) LazyTensor
y_j = LazyTensor(y[None, :, :])  # (1, N, D) LazyTensor

D_ij = ((x_i - y_j) ** 2).sum(-1).sqrt()  # Symbolic (M, N) matrix of distances
K_ij = (-D_ij).exp()  # Symbolic (M, N) Laplacian (aka. exponential) kernel matrix
a_i = K_ij @ b  # The matrix-vector product "@" can be used on "raw" PyTorch tensors!

print("a_i is now a {} of shape {}.".format(type(a_i), a_i.shape))

Note

KeOps LazyTensors have two symbolic or “virtual” axes at positions -3 and -2. Operations on the last “vector” dimension (-1) or on optional “batch” dimensions (-4 and beyond) are evaluated lazily. On the other hand, a reduction on one of the two symbolic axes (-2 or -3) triggers an explicit computation: we return a standard dense array with no symbolic axes.

Automatic differentiation

KeOps fully support the torch.autograd engine: we can backprop through KeOps reductions as easily as through vanilla PyTorch operations. For instance, coming back to the kernel dot product above, we can compute the gradient

\[g_i ~=~ \frac{\partial \sum_i \|a_i\|^2}{\partial x_i}\]

with:

[g_i] = torch.autograd.grad((a_i ** 2).sum(), [x], create_graph=True)
print("g_i is now a {} of shape {}.".format(type(g_i), g_i.shape))

As usual with PyTorch, having set the create_graph=True option allows us to compute higher-order derivatives as needed:

[h_i] = torch.autograd.grad(g_i.exp().sum(), [x], create_graph=True)
print("h_i is now a {} of shape {}.".format(type(h_i), h_i.shape))

Warning

As of today, backpropagation is not supported through the pykeops.torch.LazyTensor.min(), pykeops.torch.LazyTensor.max() or pykeops.torch.LazyTensor.Kmin() reductions: we’re working on it, but are not there just yet. Until then, a simple workaround is to use the indices computed by the pykeops.torch.LazyTensor.argmin(), pykeops.torch.LazyTensor.argmax() or pykeops.torch.LazyTensor.argKmin() reductions to define a fully differentiable PyTorch tensor as we now explain.

Coming back to our example about nearest neighbors in the unit cube:

x = torch.randn(M, 3).type(tensor)
y = torch.randn(N, 3).type(tensor)
x.requires_grad = True

x_i = LazyTensor(x[:, None, :])  # (M, 1, 3) LazyTensor
y_j = LazyTensor(y[None, :, :])  # (1, N, 3) LazyTensor
D_ij = ((x_i - y_j) ** 2).sum(-1)  # Symbolic (M, N) matrix of squared distances

We could compute the (M,) vector of squared distances to the nearest y-neighbor with:

to_nn = D_ij.min(dim=1).view(-1)

But instead, using:

s_i = D_ij.argmin(dim=1).view(-1)  # (M,) integer Torch tensor
to_nn_alt = ((x - y[s_i, :]) ** 2).sum(-1)

outputs the same result, while also allowing us to compute arbitrary gradients:

print(
    "Difference between the two vectors: {:.2e}".format((to_nn - to_nn_alt).abs().max())
)

[g_i] = torch.autograd.grad(to_nn_alt.sum(), [x])
print("g_i is now a {} of shape {}.".format(type(g_i), g_i.shape))

The only real downside here is that we had to write twice the “squared distance” formula that specifies our computation. We hope to fix this (minor) inconvenience sooner rather than later!

Batch processing

As should be expected, LazyTensors also provide full support of batch processing, with broadcasting over dummy (=1) batch dimensions:

A, B = 7, 3  # Batch dimensions

x_i = LazyTensor(torch.randn(A, B, M, 1, D))
l_i = LazyTensor(torch.randn(1, 1, M, 1, D))
y_j = LazyTensor(torch.randn(1, B, 1, N, D))
s = LazyTensor(torch.rand(A, 1, 1, 1, 1))

D_ij = ((l_i * x_i - y_j) ** 2).sum(-1)  # Symbolic (A, B, M, N, 1) LazyTensor
K_ij = -1.6 * D_ij / (1 + s ** 2)  # Some arbitrary (A, B, M, N, 1) Kernel matrix

a_i = K_ij.sum(dim=3)
print("a_i is now a {} of shape {}.".format(type(a_i), a_i.shape))

Everything works just fine, with two major caveats:

• The structure of KeOps computations is still a little bit rigid: LazyTensors should only be used in situations where the large dimensions M and N over which the main reduction is performed are in positions -3 and -2 (respectively), with vector variables in position -1 and an arbitrary number of batch dimensions beforehand. We’re working towards a full support of tensor variables, but this will probably take some time to implement and test properly…

• KeOps LazyTensors never collapse their last “dimension”, even after a .sum(-1) reduction whose keepdim argument is implicitely set to True.

print("Convenient, numpy-friendly shape:       ", K_ij.shape)
print("Actual shape, used internally by KeOps: ", K_ij._shape)

This is the reason why in the example above, a_i is a 4D Tensor of shape (7, 3, 1000, 1) and not a 3D Tensor of shape (7, 3, 1000).

Supported formulas

The full range of mathematical operations supported by LazyTensors is described in our API documentation. Let’s just mention that the lines below define valid computations:

x_i = LazyTensor(torch.randn(A, B, M, 1, D))
l_i = LazyTensor(torch.randn(1, 1, M, 1, D))
y_j = LazyTensor(torch.randn(1, B, 1, N, D))
s = LazyTensor(torch.rand(A, 1, 1, 1, 1))

F_ij = (
    (x_i ** 1.5 + y_j / l_i).cos() - (x_i | y_j) + (x_i[:, :, :, :, 2] * s.relu() * y_j)
)
print(F_ij)

a_j = F_ij.sum(dim=2)
print("a_j is now a {} of shape {}.".format(type(a_j), a_j.shape))

Enjoy! And feel free to check the next tutorial for a discussion of the varied reduction operations that can be applied to KeOps LazyTensors.