Python Matrix Factorization Evaluation

Question

I know it is so simple question but I didn't get it.

I executed the code in here and it works properly

import numpy

def matrix_factorization(R, P, Q, K, steps=5000, alpha=0.0002, beta=0.02):
    '''
    R: rating matrix
    P: |U| * K (User features matrix)
    Q: |D| * K (Item features matrix)
    K: latent features
    steps: iterations
    alpha: learning rate
    beta: regularization parameter'''
    Q = Q.T

    for step in range(steps):
        for i in range(len(R)):
            for j in range(len(R[i])):
                if R[i][j] > 0:
                    # calculate error
                    eij = R[i][j] - numpy.dot(P[i,:],Q[:,j])

                    for k in range(K):
                        # calculate gradient with a and beta parameter
                        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 range(len(R)):

            for j in range(len(R[i])):

                if R[i][j] > 0:

                    e = e + pow(R[i][j] - numpy.dot(P[i,:],Q[:,j]), 2)

                    for k in range(K):

                        e = e + (beta/2) * (pow(P[i][k],2) + pow(Q[k][j],2))
        # 0.001: local minimum
        if e < 0.001:

            break

    return P, Q.T

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)
print(nR)

I know the math behind the matrix factorization model. The point I don't understand is the test part.

Let me explain with an example.

I'm giving the R matrix like the following:

USER - MOVIE Matrix: (Input)

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

According to the matrix factorization model, my output is as follows

The output of Matrix Factorizationion:

array([[5.05061059, 2.76932235, 6.00910081, 0.99918217],
       [3.9246345 , 2.15333139, 4.77284691, 0.9999652 ],
       [1.12424323, 0.64611747, 3.53300274, 4.96832669],
       [0.94119495, 0.53978842, 2.87426388, 3.9788853 ],
       [2.71213676, 1.50889704, 4.83895101, 4.0215367 ]])

Question

Okay cool, learned with values close to the original matrix. What will this matrix do for me??

How will matrix factorization calculate the rating score that the current user will give to a new movie? or The rating a new user will give to an existing movie?

In Collaborative Filtering, we calculate the probability of how much the user will score a movie that has not been watched before and we make a recommendation according to it. How we do the same intuition in matrix factorization?

Salma Salma · Accepted Answer · 2021-01-01T15:03:06

As you said, we can see that for the existing ratings, we have the approximations very close to the true values to the R matrix, and we also get some 'predictions' of the unknown values (values with zero).

In the tutorial you sent, you can easily see that U1 and U2 have similar ratings, and they both rated D1 and D2 high, while the rest of the users preferred D3, D4, and D5.

The algorithm you used can associate the users (P) and items (Q) based on different features (K), and the predictions also follow these associations.

For example, we can see that the predicted rating of U4 on D3 is 4.59 because U4 and U5 both rated D4 high and so on for the rest of the users and items. That's why the generated matrix values are so close to the original matrix. I hope that clears things out!

Python Matrix Factorization Evaluation

1 Answers