How to compute the cosine_similarity in pytorch for all rows in a matrix with respect to all rows in another matrix
In pytorch, given that I have 2 matrixes how would I compute cosine similarity of all rows in each with all rows in the other.

For example 

Given the input =

matrix_1 = [a b] 
           [c d] 
matrix_2 = [e f] 
           [g h]


I would like the output to be 

output =

 [cosine_sim([a b] [e f])  cosine_sim([a b] [g h])]
 [cosine_sim([c d] [e f])  cosine_sim([c d] [g h])] 


At the moment I am using torch.nn.functional.cosine_similarity(matrix_1, matrix_2) which returns the cosine of the row with only that corresponding row in the other matrix. 

In my example I have only 2 rows, but I would like a solution which works for many rows. I would even like to handle the case where the number of rows in the each matrix is different. 

I realize that I could use the expand, however I want to do it without using such a large memory footprint.

By manually computing the similarity and playing with matrix multiplication + transposition:

import torch
from scipy import spatial
import numpy as np

a = torch.randn(2, 2)
b = torch.randn(3, 2) # different row number, for the fun

# Given that cos_sim(u, v) = dot(u, v) / (norm(u) * norm(v))
#                          = dot(u / norm(u), v / norm(v))
# We fist normalize the rows, before computing their dot products via transposition:
a_norm = a / a.norm(dim=1)[:, None]
b_norm = b / b.norm(dim=1)[:, None]
res = torch.mm(a_norm, b_norm.transpose(0,1))
print(res)
#  0.9978 -0.9986 -0.9985
# -0.8629  0.9172  0.9172

# -------
# Let's verify with numpy/scipy if our computations are correct:
a_n = a.numpy()
b_n = b.numpy()
res_n = np.zeros((2, 3))
for i in range(2):
    for j in range(3):
        # cos_sim(u, v) = 1 - cos_dist(u, v)
        res_n[i, j] = 1 - spatial.distance.cosine(a_n[i], b_n[j])
print(res_n)
# [[ 0.9978022  -0.99855876 -0.99854881]
#  [-0.86285472  0.91716063  0.9172349 ]]

