Advanced usage: Vi, Vj, Pm helpers and symbolic variables

This tutorial shows some advanced features of the LazyTensor class.

import time

import torch

from pykeops.torch import LazyTensor

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

The Vi, Vj, Pm decorators

This part presents an alternative style for using the KeOps LazyTensor wrapper, that some users may find more convenient. The idea is to always input 2D tensors, and use the Vi, Vj helpers described below to specify wether the tensor is to be understood as indexed by i (i.e. with an equivalent shape of the form (M,1,D)) or by j (shape of the form (1,N,D)). Note that it is currently not possible to use additional batch dimensions with this specific syntax.

Here is how it works, if we want to perform a simple gaussian convolution.

We first create the dataset using 2D tensors:

M, N = (100000, 200000) if use_cuda else (1000, 2000)
D = 3

x = torch.randn(M, D).type(tensor)
y = torch.randn(N, D).type(tensor)

Then we use Vi and Vj to convert to KeOps LazyTensor objects

from pykeops.torch import Vi, Vj

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

and perform our operations:

D2xy = ((x_i - y_j) ** 2).sum()
gamma = D2xy.sum_reduction(dim=1)

Note that in the first line we used sum without any axis or dim parameter. This is equivalent to sum(-1) or sum(dim=2), because the axis parameter is set to 2 by default. But speaking about dim=2 here with the Vi, Vj helpers could be misleading. Similarly we used sum_reduction instead of sum to make it clear that we perform a reduction, but sum and sum_reduction with dim=0 or 1 are equivalent (however sum_reduction with dim=2 is forbidden)

We have not spoken about Pm yet. In fact Pm is used to introduce scalars or 1D vectors of parameters into formulas, but it is useless in such examples because scalars, lists of scalars, 0D or 1D NumPy vectors are automatically converted into parameters when combined with KeOps formulas. We will have to use Pm in other parts below.

Other examples

All KeOps operations and reductions are available, either via operators or methods. Here are one line examples

Getting indices of closest point between x and y:

indmin = ((x_i - y_j) ** 2).sum().argmin(axis=0)

Out:

Compiling libKeOpstorch4bd7e3de28 in /home/bcharlier/tmp/libkeops/pykeops/common/../build//build-libKeOpstorch4bd7e3de28:
       formula: ArgMin_Reduction(Sum(Square((Var(0,3,0) - Var(1,3,1)))),1)
       aliases: Var(0,3,0); Var(1,3,1);
       dtype  : float32
... Done.

Scalar product, absolute value, power operator, and a SoftMax type reduction:

res = (abs(x_i | y_j) ** 1.5).sumsoftmaxweight(x_i, axis=1)

Out:

Compiling libKeOpstorch332db0cd11 in /home/bcharlier/tmp/libkeops/pykeops/common/../build//build-libKeOpstorch332db0cd11:
       formula: Max_SumShiftExpWeight_Reduction(Powf(Abs((Var(0,3,0) | Var(1,3,1))), Var(2,1,2)),0,Concat(IntCst(1),Var(0,3,0)))
       aliases: Var(0,3,0); Var(1,3,1); Var(2,1,2);
       dtype  : float32
... Done.

The [] operator can be used to do element selection or slicing (Elem or Extract operation in KeOps).

res = (x_i[:2] * y_j[2:] - x_i[2:] * y_j[:2]).sqnorm2().sum(axis=1)

Out:

Compiling libKeOpstorch5b36959de3 in /home/bcharlier/tmp/libkeops/pykeops/common/../build//build-libKeOpstorch5b36959de3:
       formula: Sum_Reduction(SqNorm2(((Extract(Var(0,3,0),0,2) * Extract(Var(1,3,1),2,1)) - (Extract(Var(0,3,0),2,1) * Extract(Var(1,3,1),0,2)))),0)
       aliases: Var(0,3,0); Var(1,3,1);
       dtype  : float32
... Done.

Kernel inversion : let’s do a gaussian kernel inversion. Note that we have to use both Vi and Vj helpers on the same tensor x here.

e_i = Vi(torch.rand(M, D).type(tensor))
x_j = Vj(x)
D2xx = LazyTensor.sum((x_i - x_j) ** 2)
sigma = 0.25
Kxx = (-D2xx / sigma ** 2).exp()
res = LazyTensor.solve(Kxx, e_i, alpha=.1)

Out:

Compiling libKeOpstorchb76a69760c in /home/bcharlier/tmp/libkeops/pykeops/common/../build//build-libKeOpstorchb76a69760c:
       formula: Sum_Reduction((Exp((Minus(Sum(Square((Var(1,3,0) - Var(2,3,1))))) / Var(3,1,2))) * Var(0,3,1)),0)
       aliases: Var(0,3,1); Var(1,3,0); Var(2,3,1); Var(3,1,2);
       dtype  : float32
... Done.

Use of loops or vector operations for sums of kernels

Let us now perform again a kernel convolution, but replacing the gaussian kernel by a sum of 4 gaussian kernels with different sigma widths. This can be done as follows with a for loop:

sigmas = tensor([0.5, 1.0, 2.0, 4.0])
b_j = Vj(torch.rand(N, D).type(tensor))
Kxy = 0
for sigma in sigmas:
    Kxy += LazyTensor.exp(-D2xy / sigma ** 2)
gamma = (Kxy * b_j).sum_reduction(axis=1)

Out:

Compiling libKeOpstorch828aafbac0 in /home/bcharlier/tmp/libkeops/pykeops/common/../build//build-libKeOpstorch828aafbac0:
       formula: Sum_Reduction(((((Exp((Minus(Sum(Square((Var(0,3,0) - Var(1,3,1))))) / Var(2,1,2))) + Exp((Minus(Sum(Square((Var(0,3,0) - Var(1,3,1))))) / Var(3,1,2)))) + Exp((Minus(Sum(Square((Var(0,3,0) - Var(1,3,1))))) / Var(4,1,2)))) + Exp((Minus(Sum(Square((Var(0,3,0) - Var(1,3,1))))) / Var(5,1,2)))) * Var(6,3,1)),0)
       aliases: Var(0,3,0); Var(1,3,1); Var(2,1,2); Var(3,1,2); Var(4,1,2); Var(5,1,2); Var(6,3,1);
       dtype  : float32
... Done.

Note again that after the for loop, no actual computation has been performed. So we can actually build formulas with much more flexibility than with the use of Genred.

Ok, this was just to showcase the use of a for loop, however in this case there is no need for a for loop, we can do simply:

Kxy = LazyTensor.exp(-D2xy / sigmas ** 2).sum()
gamma = (Kxy * b_j).sum_reduction(axis=1)

Out:

Compiling libKeOpstorchaa2c0594a5 in /home/bcharlier/tmp/libkeops/pykeops/common/../build//build-libKeOpstorchaa2c0594a5:
       formula: Sum_Reduction((Sum(Exp((Minus(Sum(Square((Var(0,3,0) - Var(1,3,1))))) / Var(2,4,2)))) * Var(3,3,1)),0)
       aliases: Var(0,3,0); Var(1,3,1); Var(2,4,2); Var(3,3,1);
       dtype  : float32
... Done.

This is because all operations are broadcasted, so the / operation above works and corresponds to a ./ (scalar-vector element-wise division)

The “no call” mode

When using a reduction operation, the user has the choice to actually not perform the computation directly and instead output a KeOps object which is direclty callable. This can be done using the “call=False” option

M, N = (100000, 200000) if use_cuda else (1000, 2000)
D, Dv = 3, 4

x = torch.randn(M, D).type(tensor)
y = torch.randn(N, D).type(tensor)
b = torch.randn(N, Dv).type(tensor)
sigmas = tensor([0.5, 1.0, 2.0, 4.0])
from pykeops.torch import Vi, Vj, Pm

x_i = Vi(x)  # (M, 1, D) LazyTensor, equivalent to LazyTensor( x[:,None,:] )
y_j = Vj(y)  # (1, N, D) LazyTensor, equivalent to LazyTensor( y[None,:,:] )
b_j = Vj(b)  # (1, N, D) LazyTensor, equivalent to LazyTensor( b[None,:,:] )

D2xy = ((x_i - y_j) ** 2).sum(-1)

Kxy = LazyTensor.exp(-D2xy / sigmas ** 2).sum()

gammafun = (Kxy * b_j).sum_reduction(axis=1, call=False)

Here gammafun is a function and can be evaluated later

gamma = gammafun()

Out:

Compiling libKeOpstorch2119901533 in /home/bcharlier/tmp/libkeops/pykeops/common/../build//build-libKeOpstorch2119901533:
       formula: Sum_Reduction((Sum(Exp((Minus(Sum(Square((Var(0,3,0) - Var(1,3,1))))) / Var(2,4,2)))) * Var(3,4,1)),0)
       aliases: Var(0,3,0); Var(1,3,1); Var(2,4,2); Var(3,4,1);
       dtype  : float32
... Done.

This is usefull in order to avoid the small overhead caused by using the container syntax inside loops if one wants to perform a large number of times the same reduction. Here is an example where we compare the two approaches on small size data to see the effect of the overhead

M, N = 50, 30
x = torch.rand(M, D).type(tensor)
y = torch.rand(N, D).type(tensor)
beta = torch.rand(N, Dv).type(tensor)

xi, yj, bj = Vi(x), Vj(y), Vj(beta)
dxy2 = LazyTensor.sum((xi - yj) ** 2)

Niter = 1000

start = time.time()
for k in range(Niter):
    Kxyb = LazyTensor.exp(-dxy2 / sigmas ** 2).sum() * bj
    gamma = Kxyb.sum_reduction(axis=1)
end = time.time()
print('Timing for {} iterations: {:.5f}s = {} x {:.5f}s'.format(
    Niter, end - start, Niter, (end - start) / Niter))

start = time.time()
Kxyb = LazyTensor.exp(-dxy2 / sigmas ** 2).sum() * bj
gammafun = Kxyb.sum_reduction(axis=1, call=False)
for k in range(Niter):
    gamma = gammafun()
end = time.time()
print('Timing for {} iterations: {:.5f}s = {} x {:.5f}s'.format(
    Niter, end - start, Niter, (end - start) / Niter))

Out:

Timing for 1000 iterations: 0.50551s = 1000 x 0.00051s
Timing for 1000 iterations: 0.35033s = 1000 x 0.00035s

Of course this means the user has to perform in-place operations over tensors x, y, beta inside the loop, otherwise the result of the call to gammafun will always be the same. This is not very convenient, so we provide also a “symbolic variables” syntax.

Using “symbolic” variables in formulas

Instead of inputing tensors to the Vi, Vj, Pm helpers, one may specify the variables as symbolic, providing an index and a dimension:

xi = Vi(0, D)
yj = Vj(1, D)
bj = Vj(2, Dv)
Sigmas = Pm(3, 4)

Now we build the formula as before

dxy2 = LazyTensor.sum((xi - yj) ** 2)
Kxyb = LazyTensor.exp(-dxy2 / Sigmas ** 2).sum() * bj
gammafun = Kxyb.sum_reduction(axis=1)

Note that we did not have to specify call=False because since the variables are symbolic, no computation can be done of course. So the ouput is automatically a function. We can evaluate it by providing the arguments in the order specified by the index argument given to Vi, Vj, Pm:

gamma = gammafun(x, y, beta, sigmas)

Out:

Compiling libKeOpstorchbc0e6568c2 in /home/bcharlier/tmp/libkeops/pykeops/common/../build//build-libKeOpstorchbc0e6568c2:
       formula: Sum_Reduction((Sum(Exp((Minus(Sum(Square((Var(0,3,0) - Var(1,3,1))))) / Square(Var(3,4,2))))) * Var(2,4,1)),0)
       aliases: Var(0,3,0); Var(1,3,1); Var(2,4,1); Var(3,4,2);
       dtype  : float32
... Done.

Symbolic and non symbolic variables can be mixed. For example if we want to fix x, beta and sigmas in the previous example and make the reduction a function of y only we can write:

xi = Vi(x)
yj = Vj(0, D)
bj = Vj(beta)

dxy2 = LazyTensor.sum((xi - yj) ** 2)
Kxyb = LazyTensor.exp(-dxy2 / sigmas ** 2).sum() * bj
gammafun = Kxyb.sum_reduction(axis=1)
print(gammafun)

gamma = gammafun(y)

Out:

KeOps LazyTensor
    formula: (Sum(Exp((Minus(Sum(Square((Var(1,3,0) - Var(0,3,1))))) / Var(2,4,2)))) * Var(3,4,1))
    symbolic variables: Var(0, 3, 1)
    shape: (50, 30, 4)
    reduction: Sum (axis=1)
Compiling libKeOpstorch33b4ddd7d3 in /home/bcharlier/tmp/libkeops/pykeops/common/../build//build-libKeOpstorch33b4ddd7d3:
       formula: Sum_Reduction((Sum(Exp((Minus(Sum(Square((Var(1,3,0) - Var(0,3,1))))) / Var(2,4,2)))) * Var(3,4,1)),0)
       aliases: Var(0,3,1); Var(1,3,0); Var(2,4,2); Var(3,4,1);
       dtype  : float32
... Done.

Total running time of the script: ( 3 minutes 1.533 seconds)

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