# How to make a matrix factorization system recommander faster?

I used and modified a matrix factorization function for a system recommender from quuxlabs.com. It works well for small sized input but when we get to large matrix it takes too much time. I was wondering if you had any idea to code it better to optimize the time it would take.

Here R is the rating matrix of a user with an item. 0 means that nothing has been predicted yet, the one we have to predict. K is the number of extrab features. P and Q are scores matrix on latent features that would help predict R as a whole.

import numpy

def matrix_factorization(R, P, Q, K, steps=5000, alpha=0.0002, beta=0.02):
Q = Q.T
for step in xrange(steps):
for i in xrange(len(R)):
for j in xrange(len(R[i])):
if R[i][j] > 0:
try:
eij = R[i][j] - numpy.dot(P[i,:],Q[:,j])
except IndexError:
print("Oops!  i = ",i,"and len(P) = ",len(P))
P[i,:]
break
for k in xrange(K):
P[i][k] = P[i][k] + alpha * (2 * eij * Q[k][j] - beta * P[i][k])
Q[k][j] = Q[k][j] + alpha * (2 * eij * P[i][k] - beta * Q[k][j])
eR = numpy.dot(P,Q)
e = 0
for i in xrange(len(R)):
for j in xrange(len(R[i])):
if R[i][j] > 0:
e = e + pow(R[i][j] - numpy.dot(P[i,:],Q[:,j]), 2)
for k in xrange(K):
e = e + (beta/2) * (pow(P[i][k],2) + pow(Q[k][j],2))
if e < 0.001:
break
return P, Q.T


It works well with the following input :

R = [
[5,3,0,1],
[4,0,0,1],
[1,1,0,5],
[1,0,0,4],
[0,1,5,4],
]

R = numpy.array(R)

N = len(R)
M = len(R[0])
K = 2

P = numpy.random.rand(N,K)
Q = numpy.random.rand(M,K)
nP, nQ = matrix_factorization(R, P, Q, K)

nR = numpy.dot(nP, nQ.T)


Yet, on real big matrix, the function took so long that I eventually cancelled it because I got tired of waiting.