How To Vectorize Multiple Matrix Multiplications In Numpy?

For a conceptual idea of what I mean, I have 2 data points: x_0 = np.array([0.6, 1.4])[:, None] x_1 = np.array([2.6, 3.4])[:, None] And a 2x2 matrix: y = np.array([[2, 2], [2, 2]]

Solution 1:

With x as the column stacked version of x_0, x_1 and so on, we can use np.einsum -

np.einsum('ji,jk,ki->i',x,y,x)

With a mix of np.einsum and matrix-multiplcation -

np.einsum('ij,ji->i',x.T.dot(y),x)

As stated earlier, x was assumed to be column-stacked, like so :

x = np.column_stack((x_0, x_1))

Runtime test -

In [236]: x = np.random.randint(0,255,(3,100000))

In [237]: y = np.random.randint(0,255,(3,3))

# Proposed in @titipata's post/comments under this post
In [238]: %timeit (x.T.dot(y)*x.T).sum(1)
100 loops, best of 3: 3.45 ms per loop

# Proposed earlier in this post
In [239]: %timeit np.einsum('ji,jk,ki->i',x,y,x)
1000 loops, best of 3: 832 µs per loop

# Proposed earlier in this post
In [240]: %timeit np.einsum('ij,ji->i',x.T.dot(y),x)
100 loops, best of 3: 2.6 ms per loop

Solution 2:

Basically, you want to do the operation (x.T).dot(A).dot(x) for all x that you have.

x_0 = np.array([0.6, 1.4])[:, None]
x_1 = np.array([2.6, 3.4])[:, None]
x = np.hstack((x_0, x_1)) # [[ 0.6  2.6], [ 1.4  3.4]]

The easy way to think about it is to do multiplication for all x_i that you have with y as

[x_i.dot(y).dot(x_i) for x_i in x.T]
>> [8.0, 72.0]

But of course this is not too efficient. However, you can do the trick where you can do dot product of x with y first and multiply back with itself and sum over column i.e. you manually do dot product. This will make the calculation much faster:

x = x.T
(x.dot(y) * x).sum(axis=1)
>> array([  8.,  72.])

Note that I transpose the matrix first because we want to multiply column of y to each row of x