SumSoftMaxWeight reduction

Using the numpy.Genred class, we show how to perform a computation specified through:

  • Its inputs:

    • \(x\), an array of size \(M\times 3\) made up of \(M\) vectors in \(\mathbb R^3\),

    • \(y\), an array of size \(N\times 3\) made up of \(N\) vectors in \(\mathbb R^3\),

    • \(b\), an array of size \(N\times 2\) made up of \(N\) vectors in \(\mathbb R^2\).

  • Its output:

    • \(c\), an array of size \(M\times 2\) made up of \(M\) vectors in \(\mathbb R^2\) such that

      \[c_i = \frac{\sum_j \exp(K(x_i,y_j))\,\cdot\,b_j }{\sum_j \exp(K(x_i,y_j))},\]

      with \(K(x_i,y_j) = \|x_i-y_j\|^2\).

Setup

Standard imports:

import time
import numpy as np
from pykeops.numpy import Genred
import matplotlib.pyplot as plt

Define our dataset:

M = 500  # Number of "i" points
N = 400  # Number of "j" points
D = 3    # Dimension of the ambient space
Dv = 2   # Dimension of the vectors

x = 2 * np.random.randn(M, D)
y = 2 * np.random.randn(N, D)
b = np.random.rand(N, Dv)

KeOps kernel

Create a new generic routine using the numpy.Genred constructor:

formula = 'SqDist(x,y)'
formula_weights = 'b'
aliases = ['x = Vi('+str(D)+')',   # First arg:  i-variable of size D
           'y = Vj('+str(D)+')',   # Second arg: j-variable of size D
           'b = Vj('+str(Dv)+')']  # Third arg:  j-variable of size Dv

softmax_op = Genred(formula, aliases, reduction_op='SumSoftMaxWeight', axis=1,
                    formula2=formula_weights)

# Dummy first call to warmup the GPU and get accurate timings:
_ = softmax_op(x, y, b)

Use our new function on arbitrary Numpy arrays:

start = time.time()
c = softmax_op(x, y, b)
print("Timing (KeOps implementation): ",round(time.time()-start,5),"s")

# compare with direct implementation
start = time.time()
cc  = np.sum( ( x[:,None,:] - y[None,:,:] ) ** 2, axis=2)
cc -= np.max(cc, axis=1)[:,None]  # Subtract the max to prevent numeric overflows
cc  = np.exp(cc)@b / np.sum( np.exp(cc), axis=1)[:,None]
print("Timing (Numpy implementation): ",round(time.time()-start,5),"s")

print("Relative error : ", (np.linalg.norm(c - cc) / np.linalg.norm(c)).item())

# Plot the results next to each other:
for i in range(Dv):
    plt.subplot(Dv, 1, i+1)
    plt.plot( c[:40,i],  '-', label='KeOps')
    plt.plot(cc[:40,i], '--', label='NumPy')
    plt.legend(loc='lower right')
plt.tight_layout() ; plt.show()
../../_images/sphx_glr_plot_test_softmax_numpy_001.png

Out:

Timing (KeOps implementation):  0.0013 s
Timing (Numpy implementation):  0.00874 s
Relative error :  2.9060225500263333e-16
/home/bcharlier/keops/pykeops/examples/numpy/plot_test_softmax_numpy.py:94: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
  plt.tight_layout() ; plt.show()

Total running time of the script: ( 0 minutes 9.338 seconds)

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