numpy - What's the fastest way in Python to calculate cosine similarity given sparse matrix data?

Question

Given a sparse matrix listing, what's the best way to calculate the cosine similarity between each of the columns (or rows) in the matrix? I would rather not iterate n-choose-two times.

Say the input matrix is:

A= 
[0 1 0 0 1
 0 0 1 1 1
 1 1 0 1 0]

The sparse representation is:

A = 
0, 1
0, 4
1, 2
1, 3
1, 4
2, 0
2, 1
2, 3

In Python, it's straightforward to work with the matrix-input format:

import numpy as np
from sklearn.metrics import pairwise_distances
from scipy.spatial.distance import cosine

A = np.array(
[[0, 1, 0, 0, 1],
[0, 0, 1, 1, 1],
[1, 1, 0, 1, 0]])

dist_out = 1-pairwise_distances(A, metric="cosine")
dist_out

Gives:

array([[ 1.        ,  0.40824829,  0.40824829],
       [ 0.40824829,  1.        ,  0.33333333],
       [ 0.40824829,  0.33333333,  1.        ]])

That's fine for a full-matrix input, but I really want to start with the sparse representation (due to the size and sparsity of my matrix). Any ideas about how this could best be accomplished? Thanks in advance.

Answer 1

You can compute pairwise cosine similarity on the rows of a sparse matrix directly using sklearn. As of version 0.17 it also supports sparse output:

from sklearn.metrics.pairwise import cosine_similarity
from scipy import sparse

A =  np.array([[0, 1, 0, 0, 1], [0, 0, 1, 1, 1],[1, 1, 0, 1, 0]])
A_sparse = sparse.csr_matrix(A)

similarities = cosine_similarity(A_sparse)
print('pairwise dense output:
 {}
'.format(similarities))

#also can output sparse matrices
similarities_sparse = cosine_similarity(A_sparse,dense_output=False)
print('pairwise sparse output:
 {}
'.format(similarities_sparse))

Results:

pairwise dense output:
[[ 1.          0.40824829  0.40824829]
[ 0.40824829  1.          0.33333333]
[ 0.40824829  0.33333333  1.        ]]

pairwise sparse output:
(0, 1)  0.408248290464
(0, 2)  0.408248290464
(0, 0)  1.0
(1, 0)  0.408248290464
(1, 2)  0.333333333333
(1, 1)  1.0
(2, 1)  0.333333333333
(2, 0)  0.408248290464
(2, 2)  1.0

If you want column-wise cosine similarities simply transpose your input matrix beforehand:

A_sparse.transpose()