KernelSolve reduction (with LazyTensors)

Let’s see how to solve discrete deconvolution problems using the conjugate gradient solver provided by the pykeops.torch.LazyTensor.solve() method of KeOps pykeops.torch.LazyTensor.

Setup

Standard imports:

import time

import torch
from matplotlib import pyplot as plt

from pykeops.torch import LazyTensor as keops
from pykeops.torch import Vi, Vj

Define our dataset:

N = 5000 if torch.cuda.is_available() else 500  # Number of points
D = 2  # Dimension of the ambient space
Dv = 2  # Dimension of the vectors (= number of linear problems to solve)
sigma = .1  # Radius of our RBF kernel

x = torch.rand(N, D, requires_grad=True)
b = torch.rand(N, Dv)
g = torch.Tensor([.5 / sigma ** 2])  # Parameter of the Gaussian RBF kernel
alpha = 0.01  # ridge regularization

Note

This operator uses a conjugate gradient solver and assumes that formula defines a symmetric, positive and definite linear reduction with respect to the alias "b" specified trough the third argument.

Apply our solver on arbitrary point clouds:

print("Solving a Gaussian linear system, with {} points in dimension {}.".format(N, D))
start = time.time()
K_xx = keops.exp(-keops.sum((Vi(x) - Vj(x)) ** 2, dim=2) / (2 * sigma ** 2))
cfun = keops.solve(K_xx, Vi(b), alpha=alpha, call=False)
c = cfun()
end = time.time()
print('Timing (KeOps implementation):', round(end - start, 5), 's')

Out:

Solving a Gaussian linear system, with 5000 points in dimension 2.
Compiling libKeOpstorch53151ee29e in /home/bcharlier/tmp/libkeops/pykeops/common/../build//build-libKeOpstorch53151ee29e:
       formula: Sum_Reduction((Exp((Minus(Sum(Square((Var(1,2,0) - Var(2,2,1))))) / Var(3,1,2))) * Var(0,2,1)),0)
       aliases: Var(0,2,1); Var(1,2,0); Var(2,2,1); Var(3,1,2);
       dtype  : float32
... Done.
Timing (KeOps implementation): 19.79931 s

Compare with a straightforward PyTorch implementation:

start = time.time()
K_xx = alpha * torch.eye(N) + torch.exp(-torch.sum((x[:, None, :] - x[None, :, :]) ** 2, dim=2) / (2 * sigma ** 2))
c_py = torch.solve(b, K_xx)[0]
end = time.time()
print('Timing (PyTorch implementation):', round(end - start, 5), 's')
print("Relative error = ", (torch.norm(c - c_py) / torch.norm(c_py)).item())

# Plot the results next to each other:
for i in range(Dv):
    plt.subplot(Dv, 1, i + 1)
    plt.plot(c.cpu().detach().numpy()[:40, i], '-', label='KeOps')
    plt.plot(c_py.cpu().detach().numpy()[:40, i], '--', label='PyTorch')
    plt.legend(loc='lower right')
plt.tight_layout();
plt.show()
../../_images/sphx_glr_plot_test_invkernel_torch_helper_001.png

Out:

Timing (PyTorch implementation): 0.38662 s
Relative error =  0.0003432204539421946
/home/bcharlier/tmp/libkeops/pykeops/examples/pytorch/plot_test_invkernel_torch_helper.py:75: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
  plt.show()

Compare the derivatives:

print(cfun.callfun)

print("1st order derivative")
e = torch.randn(N, D)
start = time.time()
u, = torch.autograd.grad(c, x, e)
end = time.time()
print('Timing (KeOps derivative):', round(end - start, 5), 's')
start = time.time()
u_py, = torch.autograd.grad(c_py, x, e)
end = time.time()
print('Timing (PyTorch derivative):', round(end - start, 5), 's')
print("Relative error = ", (torch.norm(u - u_py) / torch.norm(u_py)).item())

# Plot the results next to each other:
for i in range(Dv):
    plt.subplot(Dv, 1, i + 1)
    plt.plot(u.cpu().detach().numpy()[:40, i], '-', label='KeOps')
    plt.plot(u_py.cpu().detach().numpy()[:40, i], '--', label='PyTorch')
    plt.legend(loc='lower right')
plt.tight_layout();
plt.show()
../../_images/sphx_glr_plot_test_invkernel_torch_helper_002.png

Out:

<pykeops.torch.operations.KernelSolve object at 0x7fdb2aa2a780>
1st order derivative
Compiling libKeOpstorch1d68529d13 in /home/bcharlier/tmp/libkeops/pykeops/common/../build//build-libKeOpstorch1d68529d13:
       formula: Grad_WithSavedForward(Sum_Reduction((Exp((Minus(Sum(Square((Var(1,2,0) - Var(2,2,1))))) / Var(3,1,2))) * Var(0,2,1)),0), Var(1,2,0), Var(4,2,0), Var(5,2,0))
       aliases: Var(0,2,1); Var(1,2,0); Var(2,2,1); Var(3,1,2); Var(4,2,0); Var(5,2,0);
       dtype  : float32
... Done.
Compiling libKeOpstorchf4b4ccd9ea in /home/bcharlier/tmp/libkeops/pykeops/common/../build//build-libKeOpstorchf4b4ccd9ea:
       formula: Grad_WithSavedForward(Sum_Reduction((Exp((Minus(Sum(Square((Var(1,2,0) - Var(2,2,1))))) / Var(3,1,2))) * Var(0,2,1)),0), Var(2,2,1), Var(4,2,0), Var(5,2,0))
       aliases: Var(0,2,1); Var(1,2,0); Var(2,2,1); Var(3,1,2); Var(4,2,0); Var(5,2,0);
       dtype  : float32
... Done.
Timing (KeOps derivative): 39.20557 s
Timing (PyTorch derivative): 0.48997 s
Relative error =  0.002209658734500408
/home/bcharlier/tmp/libkeops/pykeops/examples/pytorch/plot_test_invkernel_torch_helper.py:102: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
  plt.show()

Total running time of the script: ( 1 minutes 20.878 seconds)

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