Vectorized way of calculating row-wise dot product two matrices with Scipy

Question

I want to calculate the row-wise dot product of two matrices of the same dimension as fast as possible. This is the way I am doing it:

import numpy as np
a = np.array([[1,2,3], [3,4,5]])
b = np.array([[1,2,3], [1,2,3]])
result = np.array([])
for row1, row2 in a, b:
    result = np.append(result, np.dot(row1, row2))
print result

and of course the output is:

[ 26.  14.]

Solution 1

Check out numpy.einsum for another method:

In [52]: a
Out[52]: 
array([[1, 2, 3],
       [3, 4, 5]])

In [53]: b
Out[53]: 
array([[1, 2, 3],
       [1, 2, 3]])

In [54]: einsum('ij,ij->i', a, b)
Out[54]: array([14, 26])

Looks like einsum is a bit faster than inner1d:

In [94]: %timeit inner1d(a,b)
1000000 loops, best of 3: 1.8 us per loop

In [95]: %timeit einsum('ij,ij->i', a, b)
1000000 loops, best of 3: 1.6 us per loop

In [96]: a = random.randn(10, 100)

In [97]: b = random.randn(10, 100)

In [98]: %timeit inner1d(a,b)
100000 loops, best of 3: 2.89 us per loop

In [99]: %timeit einsum('ij,ij->i', a, b)
100000 loops, best of 3: 2.03 us per loop

---

Note: NumPy is constantly evolving and improving; the relative performance of the functions shown above has probably changed over the years. If performance is important to you, run your own tests with the version of NumPy that you will be using.

Solution 2

Straightforward way to do that is:

import numpy as np
a=np.array([[1,2,3],[3,4,5]])
b=np.array([[1,2,3],[1,2,3]])
np.sum(a*b, axis=1)

which avoids the python loop and is faster in cases like:

def npsumdot(x, y):
    return np.sum(x*y, axis=1)

def loopdot(x, y):
    result = np.empty((x.shape[0]))
    for i in range(x.shape[0]):
        result[i] = np.dot(x[i], y[i])
    return result

timeit npsumdot(np.random.rand(500000,50),np.random.rand(500000,50))
# 1 loops, best of 3: 861 ms per loop
timeit loopdot(np.random.rand(500000,50),np.random.rand(500000,50))
# 1 loops, best of 3: 1.58 s per loop

Solution 3

Played around with this and found inner1d the fastest. That function however is internal, so a more robust approach is to use

numpy.einsum("ij,ij->i", a, b)

Even better is to align your memory such that the summation happens in the first dimension, e.g.,

a = numpy.random.rand(3, n)
b = numpy.random.rand(3, n)

numpy.einsum("ij,ij->j", a, b)

For 10 ** 3 <= n <= 10 ** 6, this is the fastest method, and up to twice as fast as its untransposed equivalent. The maximum occurs when the level-2 cache is maxed out, at about 2 * 10 ** 4.

Note also that the transposed summation is much faster than its untransposed equivalent.

The plot was created with perfplot (a small project of mine)

import numpy
from numpy.core.umath_tests import inner1d
import perfplot


def setup(n):
    a = numpy.random.rand(n, 3)
    b = numpy.random.rand(n, 3)
    aT = numpy.ascontiguousarray(a.T)
    bT = numpy.ascontiguousarray(b.T)
    return (a, b), (aT, bT)


b = perfplot.bench(
    setup=setup,
    n_range=[2 ** k for k in range(1, 25)],
    kernels=[
        lambda data: numpy.sum(data[0][0] * data[0][1], axis=1),
        lambda data: numpy.einsum("ij, ij->i", data[0][0], data[0][1]),
        lambda data: numpy.sum(data[1][0] * data[1][1], axis=0),
        lambda data: numpy.einsum("ij, ij->j", data[1][0], data[1][1]),
        lambda data: inner1d(data[0][0], data[0][1]),
    ],
    labels=["sum", "einsum", "sum.T", "einsum.T", "inner1d"],
    xlabel="len(a), len(b)",
)

b.save("out1.png")
b.save("out2.png", relative_to=3)

Solution 4

You'll do better avoiding the append, but I can't think of a way to avoid the python loop. A custom Ufunc perhaps? I don't think numpy.vectorize will help you here.

import numpy as np
a=np.array([[1,2,3],[3,4,5]])
b=np.array([[1,2,3],[1,2,3]])
result=np.empty((2,))
for i in range(2):
    result[i] = np.dot(a[i],b[i]))
print result

EDIT

Based on this answer, it looks like inner1d might work if the vectors in your real-world problem are 1D.

from numpy.core.umath_tests import inner1d
inner1d(a,b)  # array([14, 26])

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