# 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() Out:

Timing (KeOps implementation):  0.00079 s
Timing (Numpy implementation):  0.00969 s
Relative error :  6.615871523305019e-16


Total running time of the script: ( 0 minutes 0.144 seconds)

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