numpy Broadcasting

Broadcasting is a powerful mechanism that allows numpy to work with arrays of different shapes when performing arithmetic operations. Frequently we have a smaller array and a larger array, and we want to use the smaller array multiple times to perform some operation on the larger array.

For example, suppose that we want to add a constant vector to each row of a matrix. We could do it like this:

import numpy as np# We will add the vector v to each row of the matrix x,# storing the result in the matrix yx = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])v = np.array([1, 0, 1])y = np.empty_like(x)   # Create an empty matrix with the same shape as x# Add the vector v to each row of the matrix x with an explicit loopfor i in range(4):    y[i, :] = x[i, :] + v# Now y is the following# [[ 2  2  4]#  [ 5  5  7]#  [ 8  8 10]#  [11 11 13]]print(y)

This works; however when the matrix x is very large, computing an explicit loop in Python could be slow. Note that adding the vector v to each row of the matrix x is equivalent to forming a matrix vv by stacking multiple copies of v vertically, then performing elementwise summation of x and vv. We could implement this approach like this:

import numpy as np# We will add the vector v to each row of the matrix x,# storing the result in the matrix yx = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])v = np.array([1, 0, 1])vv = np.tile(v, (4, 1))   # Stack 4 copies of v on top of each otherprint(vv)                 # Prints "[[1 0 1]                          #          [1 0 1]                          #          [1 0 1]                          #          [1 0 1]]"y = x + vv  # Add x and vv elementwiseprint(y)  # Prints "[[ 2  2  4          #          [ 5  5  7]          #          [ 8  8 10]          #          [11 11 13]]"

Numpy broadcasting allows us to perform this computation without actually creating multiple copies of v. Consider this version, using broadcasting:

import numpy as np# We will add the vector v to each row of the matrix x,# storing the result in the matrix yx = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])v = np.array([1, 0, 1])y = x + v  # Add v to each row of x using broadcastingprint(y)  # Prints "[[ 2  2  4]          #          [ 5  5  7]          #          [ 8  8 10]          #          [11 11 13]]"

The line y = x + v works even though x has shape (4, 3) and v has shape (3,) due to broadcasting; this line works as if v actually had shape (4, 3), where each row was a copy of v, and the sum was performed elementwise.

Broadcasting two arrays together follows these rules:

1. If the arrays do not have the same rank, prepend the shape of the lower rank array with 1s until both shapes have the same length.
2. The two arrays are said to be compatible in a dimension if they have the same size in the dimension, or if one of the arrays has size 1 in that dimension.
3. The arrays can be broadcast together if they are compatible in all dimensions.
4. After broadcasting, each array behaves as if it had shape equal to the elementwise maximum of shapes of the two input arrays.
5. In any dimension where one array had size 1 and the other array had size greater than 1, the first array behaves as if it were copied along that dimension

If this explanation does not make sense, try reading the explanation from the documentation or this explanation.

Functions that support broadcasting are known as universal functions. You can find the list of all universal functions in the documentation.

Here are some applications of broadcasting:

import numpy as np# Compute outer product of vectorsv = np.array([1,2,3])  # v has shape (3,)w = np.array([4,5])    # w has shape (2,)# To compute an outer product, we first reshape v to be a column# vector of shape (3, 1); we can then broadcast it against w to yield# an output of shape (3, 2), which is the outer product of v and w:# [[ 4  5]#  [ 8 10]#  [12 15]]print(np.reshape(v, (3, 1)) * w)# Add a vector to each row of a matrixx = np.array([[1,2,3], [4,5,6]])# x has shape (2, 3) and v has shape (3,) so they broadcast to (2, 3),# giving the following matrix:# [[2 4 6]#  [5 7 9]]print(x + v)# Add a vector to each column of a matrix# x has shape (2, 3) and w has shape (2,).# If we transpose x then it has shape (3, 2) and can be broadcast# against w to yield a result of shape (3, 2); transposing this result# yields the final result of shape (2, 3) which is the matrix x with# the vector w added to each column. Gives the following matrix:# [[ 5  6  7]#  [ 9 10 11]]print((x.T + w).T)# Another solution is to reshape w to be a column vector of shape (2, 1);# we can then broadcast it directly against x to produce the same# output.print(x + np.reshape(w, (2, 1)))# Multiply a matrix by a constant:# x has shape (2, 3). Numpy treats scalars as arrays of shape ();# these can be broadcast together to shape (2, 3), producing the# following array:# [[ 2  4  6]#  [ 8 10 12]]print(x * 2)

Broadcasting typically makes your code more concise and faster, so you should strive to use it where possible.

Numpy Documentation

This brief overview has touched on many of the important things that you need to know about numpy, but is far from complete. Check out the numpy reference to find out much more about numpy.